Is it overkill? Probably.
Schools generally do a good job maintaining their aircraft (they're used hard, but kept airworthy because if they're out of service the school is losing money). Most of what you should be concerned about is stuff you should find on a thorough preflight. You would need some specialist knowledge to determine if a log entry is a "problem" or not (absent something like "repaired right main gear attach point and installed new right main landing gear assembly" showing up).
Most of the schools/rental outfits around here keep a status sheet with the aircraft's dispatch documents so you (and they) can easily see when required time-in-service or calendar-time inspections are coming due. If there isn't one I'd make a point of asking about inspections and such to be sure they've been done, but otherwise I'd be pretty trusting.
Bear in mind that there's nothing stopping an unscrupulous school or mechanic from "pencil-whipping" inspections. An aircraft can be perfectly airworthy on paper but full of "How could they possibly have missed that?!" maintenance discrepancies when you preflight it.
Should they let you review the logs with an instructor or mechanic if you ask? I think so.
If you want to do this you should let them know in advance so they can have the logs available.
Bottom line, as PIC you need to be sure that the aircraft is in an airworthy condition (which includes compliance with required inspections, ADs, etc.) and if you feel you need to see the logs to verify that it's your call.
Reviewing the logs probably isn't a common request for casual rentals, but particularly if you're taking flight lessons there spending some time and reviewing the maintenance records for an aircraft is a great learning experience.