"Ms. Hopper is awesome and will do great at your university, except I really don't want her to go there, I want her to stay with meeeeeeeee."Sometimes, the clingy professors will trash the student too, sort of like in the way I tell people, "this chocolate cake is TERRIBLE, you definitely don't want any."
While it's true good students are hard to find, and showing some level of adoration and commitment toward them is a good idea, clinging too tightly is bad practice professionally and managerially. And in fact, I've heard some stories of clingy advisors that cross me as borderline abusive.
Maybe in other fields this sort of thing is tolerable, but a graduate student in CS, even a bad one, can get a job anywhere, and make 5 times as much as they would as a PhD student. So it doesn't cross me as particularly clever to treat them poorly.