I think this is all part of Quirrel's long game. Voldemort could have taken Magical Britain by force, but didn't. Why Voldemort, then? To be a scary villain and make people rally to "David Munroe". Why? Political power is one answer, and it has been much talked about in the story, but I'm not convinced. "Rule Magical Britain" seems too small. But Magical Britain is the site of Merlin's creations. It contains the Philosopher's Stone. It contains Nicolas Flamel and his lore. It contains the Department of Mysteries, the Ministry has control over time-turners, and so on. Tom Riddle has been playing a very long game to obtain magical power. He has the secrets of Slytherin. He isn't stopping there. Because of the Interdict of Merlin and the resulting decline of magic, old magic is powerful magic. This makes Flamel (Baba Yaga?) a huge, obvious, and dangerous target for Riddle. Dumbledore is as powerful as he is in large part because of what Flamel has shared with him. Riddle wants that power too, and force will probably not do the trick. What he needs is to inhabit a personality that Flamel and/or Dumbledore would support during a time of great need.

The Philosopher's Stone is a huge milestone on his journey to power, but it's not the end. I think Quirrel is setting up a narrative in which Harry destroys Voldemort/Quirrel, but Riddle actually ends up inhabiting Harry (or already does). "Voldemort" can, of course, be resurrected as a convenient antagonist at any time, creating a crisis that will induce Dumbledore to teach Harry all he knows. Dumbledore can be killed, in either that or a subsequent crisis, inducing Flamel to share with Harry even more than Dumbledore knew. And Harry, as leader of Magical Britain and potentially Head Warlock, etc., will have access to other powers as well.

So that's why Harry has to be along, and possibly why Quirrel has suddenly turned obviously villainous instead of just taking Harry up on his offer to help get the stone. Harry is part of the narrative; he has to be there so he can be the hero. Harry's remaining hour on his time-turner might be an intended part of this to make the whole thing plausible to Dumbledore and/or the rest of Magical Britain, who must believe in the end that Harry did actually best Voldemort in some clever way.