I'm not a professor neither, but a previous grad student who had very loose/inexistent supervision. Here are my thoughts on the subject. I know that loose supervision of PhD students is popular on this site (see the frequently upvoted comment here), but I think caution is necessary. PhD students start grad school with various degrees of maturity and supervision should be tailored to every individual.
Dear professors, PhD students are your Padawans. That is, they should certainly not have to live in your shadow, but they probably do not fully know how to invest their efforts appropriately. Kindly telling them that, if they wish to pursue an academic career focusing the efforts on a subject is the way to go, is going to be valuable. Let them be free to follow your advice or not, but they'll know what to expect.
It also depends on the field I guess, but in mine (metrology/biomechanics), the chances of achieving good research worthy of publication in a good journal on your own are practically nonexistent. Even if they do manage to publish, if it's in several different fields, people will wonder what is their primary field of research, see the discussion on this thread.
Now, don't get me wrong, I'm not saying students have to be babysat, they should have to show that they can conduct research independently, that is the point (in my opinion) of pursuing a PhD degree. But if they do so with guidance, they will hopefully be independent and attractive for hiring committees.
What do you mean by independent research?
To me it means being able to do the following without the supervisor having to give instructions:
- see a gap or a boundary in the current knowledge
- come up with a plan to try to answer/expand it
- ask for/marshal the necessary resources
- do the experiments appropriately
- know where and how to publish the results.