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The reputation of Z (whatever that is) may actually have an effect on whether you get into a top grad school.

I knew a professor who said if they knew nothing about or had never heard professionally about any work done by the professors or graduates of the applicant's undergraduate department, they would usually reject that applicant, marking it, "insufficient preparation."

That sounds harsh, but he also said, "Every so often we'd make an exception, and it always turned out to be a mistake."

(Also, if you write the sentences like, "I applied to MIT for the sake of completeness," it does sound like MIT would be a good cultural fit.)