Eliezer may have altered the requirements to make achieving the same goal more difficult for Quirrelmort, because if he wanted to he could easily have already gathered all the ingredients the original Voldemort did, and probably without anyone noticing to boot.

First, ask yourself how you would set things up if you were Lord Voldemort. Then, reread Ch. 53.

I haven't exhaustively read all these comments and all the reviews, so these theories might have been presented before, but here are two things I'm thinking:

-1- (medium probability) Quirrell seemed genuinely surprised to learn Harry had a mysterious dark side after the learning to lose chapter. He may not have realized this until they met, but he probably generated the hypothesis that it was a fragment of Voldemort very quickly. I think his bringing a Dementor to Hogwarts was his attempt to verify this hypothesis.

It was mentioned in Stanford Prison Experiment both that Quirrell is unusually sensitive to Dementors (probably because Voldy is unusually afraid of death) and that Harry's dark side is unusually sensitive to Dementors. Since Quirrell knows he is unusually sensitive, he theorizes that if Harry's dark side is really Voldy, Harry should be unusually sensitive. So he brings in the Dementor, arranges for the wand to be placed next to it, waits to see how badly Harry is harmed, and then pulls the wand away before there's any permanent damage.

Because of how badly Harry was harmed, he concludes that he probably does have a fragment of Voldemort in him. He asks Harry where he would hide horcruxes as a confirmation of this.

-2- (loooong stretch, low probability) We know that unspecified bad things happen if you meet yourself time-turning and that this is generally discouraged. If Voldemort went back in time at some point (maybe after his defeat in canon) thus causing some of the differences between canon and MoR, then the sense-of-doom interaction between his and Harry's magic could be Voldemort meeting Voldemort-fragment out of temporal sync with itself.