If I understand correctly, the OP's question is mainly:
"Why would a researcher support PhD students without the benefit of writing papers with them?"
(Where "support" can mean a number of things, such as hiring and supervising students in the first place, "allowing" them to publish on their own, encouraging them to publish on their own, and so on.)
Disclaimer: As mentioned in a comment, I am not completely sure whether I interpret the focus of the OP's question correctly, so let me specify that this answer refers to the presumed question that I marked in bold above.
A lot of important things have already been mentioned in other answers and comments (in particular, the dependency on the field), but I think in order to specifically answer the question, it is important to point out the following observation:
The question is based on the premise that the only primary goal of a researcher is to maximize their paper output (where "paper output" is probably a - somewhat individual - function of both qualitiy and quanitity, and in some fields also of the position within the list of authors).
However, this premise is incorrect.
The actions of human beings (rumour has it that even some scientists are human beings...) depend on various intrensic and extrinsic motivations and - most importantly - on the very complex mixture and interplay between the both.
Here is a non-comprehensive lists of goals that, say, a professor at a research university might (or sometimes might not) seek to achieve (in arbitrary order):
Make for a living, i.e., earn sufficient (what ever this means) money and/or retain a high amount of job security.
Gain and retain social esteem (including self-esteem, esteem by their family and friends, by colleagues and collaborators, their students, their university, their wider scientific community, sometimes even by complete strangers, and so on).
Enhance scientific progress in their specific subfield(s).
Promote their subfield(s) and, in particular, those parts/theories within their subfield(s) that they are particularly fond of (for instance, because they think a theory is particularly important, or because they consider it "their" theory, "their baby", "their legacy", or whatever.)
Make a contribution to solving problems that human societies are facing.
Have fun during their work.
Develop and study exciting ideas.
Collaborate with other people (most humans like social interactions, at least to a certain extent, on collaborations are one kind of social interaction).
Retain sufficient spare time to spend with their family, their hobbies, etc.
Behave what they consider ethical.
Co-authoring papers is not an end on itself; instead, it can be seen as one of several means that are instrumental to pursue some of the goals listed above (and probably also further goals). In fact, they are certainly a rather important mean, but there are also other things that can - and should - be done in order to achieve goals such as the ones listed above.
One of these further means is supervising PhD students, even if the supervision does not result in additional papers on the supervisors CV.
So to actually answer the question, let me discuss by a few examples how supervising students can help to achieve the goals mentioned above:
(The order is, again, irrelevant; I enumerate the points merely to facilitate later reference in case that some users wish to comment.)
(1) An important part of "making for a living" is to get a permanent position first (for instance, in the US this is what it typically means to "get tenure").
In order to get such a position, co-authoring many papers is certainly helpful (at least at the paper's quality does not significantly suffer from their numbers), but hiring committees, tenure committees, etc., will also expect other things, for instance, experience in the supervision of PhD students. So supervising PhD students can mean a considerable benefit for the superviser, even of the latter does not co-author any (or not many) papers of the students.
For people who already have permanent positions, the pressure is probably a bit less; but still it is in many cases considered as part of their job to supervise PhD students. If they don't do this, this might (or might not, depending on the circumstances) cause them considerable problems with their employer (i.e., their university).
(2) Supervising PhD students will often earn people social esteem from several sources: for instance, from the PhD students themselves (at least if the supervisor does a decent job), from their department, from other colleagues. Depending on the culture of the scientific field, the country, the university and even the department, the amount of esteem that a supervisor earns might even be considerably higher of their PhD students are perceived as working quite independently.
On the other end of the spectrum, if you often claim co-authorship for contributions which are usually not considered sufficient for co-authorship within your field, precisely the opposite thing is likely to happen: You might earn contempt instead of esteem.
(3) An excellent PhD student will obviously be an asset for scientific progress, so if the supervisor has an execellent PhD student, they can befinit scientific progress by supporting this student. Giving excellent students the opportunity to work independently can benefit these students in various ways (and might hurt them in others, of course - it certainly depends on the situation and the people involved).
(4) If a PhD student specialises in the same (or a closely related) theory as their supervisor, then it will likely help to promote this theory if the student successfully pursues an academic career after their PhD. In many (though not all) cases, such a career will be supported if the student has evidence of their ability to work indepently - and papers authored by the student alone are sometimes (often?) considered as such evidence. Similary, papers that the student wrote with other collaborators, but not with their supervisor as a co-author, have a good to chance to be interpreted in the sense that "the student's ability to do research is not ultimately tied to their supervisor", which can also benefit their career opportunities.
(5) In order to help solving society's problems by means of scientific research, it is a reasonable course of action to choose those PhD candidates who are most likely to do an excellent job and make good contributions, rather than to choose those who are most likely to co-author papers with you.
Also, leaving people a good amount of freedom and giving them the opportunity to work more independently, will often increase the probability for good new ideas and innovation.
(6) People are more likely to like their work if there's a good climate at their lab. Professors who prevent students from publishing papers on their own even if the students were perfectly up to it, are not likely to have a positive impact that climate. Professors who force their name onto articles to which they haven't contributed at all, are also unlikely to have a positive impact on that climate (at least be this particular action; of course, they might still impact the climate positively by other things they do).
So if a professor is does such things too often, there's a non-negligible chance that this turns (or at least takes part in turning) their own lab into a place where they wouldn't like to work and where their students wouldn't like to work, either. The first of those consequence clearly has a direct negative impact on the professor's well being, while the second can have negative impacts on some others of the proefessor's possible goals discussed here (for instance, to enhance scientific knowledge, since dissatisfied students are less likely to do good work).
(7) Developing exciting ideas is more likely if you work with people who are able to think and work independently. But there is certainly a positive correlation between people's ability to think and work indepedently and their desire to actually do think and work indepdently. So I you like to work with students in order to develop exciting ideas, you're often better off if you supervise students you are able to think and work independently - but this will sometimes (often?) have the consequence, that those students will also expect the possibility to work independently to a certain extent.
(8) If you like collaborating with other people, then you do, of course, have an incentive to... well, collaborate with other people. This particular incentive is independent of the question whether this collaboration also yields an additional paper on your CV. (But surely, in many cases the prospect to get an additional paper on your CV might be another and additional incentive for a collaboration.)
Whether a collaboration will probably result in a paper with you as co-author, depends, of course, on many things. If you supervise a PhD student, then your contribution might sometimes (though certainly not always) be of a kind which does not automatically warrant you co-authorship in some (!) scientific fields.
(9) If you have a lot of things to do (as professors typically have), and would still like to reserve non-zero spare time, then it is reasonable to delegate a certain amount of work. For instance, if you feel that a PhD student is up to doing a certain project mostly on their own, and that they will probably benefit from doing so, then you might choose to indeed let them do it mostly on their own.
This will (again, often, though probably not always) reduce you're committment in terms of working hours you put into the project, since giving guidance and feedback is less time consuming than making more detailed contributions to the project. On the other hand, this might result in a paper for which you don't have a good claim to be a co-author.
(10) Many (not all, obviously) people prefer to keep a clean conscience - so they try not to engage in actions which they consider unethical. Some people (their percentage is certainly field dependent) do consider it unethical to write their name on a paper they contributed (almost) nothing to.
Please note that this is just a sample list (which is light years from being comprehensive); I wrote it in order to demonstrate two things:
Researcher do have many incentives to supervise students, even if this doesn't always yield a new paper for the supervisor.
In some situations, insisting on co-authorship might even be counterproductive to some (or many) of the professor's goals.
Final remarks.
Please also keep in mind that, while many people have a considerable interest in performing very well on their jobs and many people put a very large amount of work into achieving this, most of them have little interest in over-optimising their performance at the upper end ot the scale.
For instance, assume you are an accomplished researcher and tenured professor, with a lot of experience and many achievements; you write many high-quality papers, are liked by your scientific community, do really enjoy your job, and like to work with your students. Then there is only very little incentive to insist on yet one more paper on your CV.
Say, you're in such a situation and you're facing a "borderline case" of a project by one of your students, where one could well argue for or against your co-authorship, and both decisions would be unlikely to have any considerable negative impacts on anyone: in such a situation, why would you really care whether you're going to have this paper on your CV or not?
One more word on field dependence: What has, as far as I can see, not been emphasised in this discussion so so far, is that fields do also have different conventions on how to order the authors of a paper. For instance, in pure math, authors are almost exclusively ordered alphabetically. In more applied fields, on the other hand, authors are typically ordered to indicate how much they contributed to the paper.
In a field where authors are ordered by contributions, there's a natural option to include authors who contributed significantly less to the paper than other authors. In a field where authors are ordered alphabetically, though, the minimal contribution in order to be recognised as a co-author will naturally be higher.