As with several other questions on this site, my suggestion is to flip your question around, and ask yourself "What would the department gain by admitting those students?"
Given your suggestion that these students pay their own way, presumably your answer is "Money". But I'd suggest to you that PhD students are an intensely inefficient way to generate income for a department for a number of reasons:
- PhD classes are small. At my former institution, even for the classes that were required for PhD students, the class size was easily dwarfed by most undergraduate classes. The ratio of students paying tuition to instructors being paid to teach isn't particularly favorable.
- This ratio becomes worse when, as @DmitrySavostyanov points out, you factor in that PhD students, to be successful, require more than just admission. They need a faculty advisor, and that faculty advisor has to actually do things, rather than just sign off on your course plan for the year. They'll need a committee. They may or may not need resources to actually work on their projects, and these resources are considerably above what might be needed for an undergraduate.
- It also becomes worse when you keep in mind that at many institutions, tuition scales by the number of classes you take. So as PhD students enter the "post-coursework" period of their career (and the resources they demand increases) they're actually paying less tuition, not more.
Beyond that, as some people have noted, failing to get a funded position anywhere is a pretty strong signaling mechanism that, regardless of how much they might think this is a good idea, the applicant may want to consider other options. Departments care about their reputation - both for ego reasons, and very practical reasons like "This effects our faculty and student's job prospects, funding opportunities, etc." Following a path where you admit tons of students who have a higher probability of being unqualified risks being thought of as a "diploma mill", and that's...pretty much a disaster for a department's reputation.
There's also simple self-interest - while in the short term flooding the pipeline with PhD students might be somewhat beneficial, in the long term there's no benefit to having a number of graduates well above equilibrium.
It should be noted however that in several fields, Masters degrees, which are much less resource intensive than PhDs, are indeed used as income generating programs.