Expired inspection stickers on rental cars aren’t that rare. I’ve known at least three people who were ticketed for expired Texas State Inspections while driving a rental car. Apparently the rental companies don’t think to check this. Technically speaking, they aren’t responsible for the ticket, fines and increase auto insurance rates (in Texas this is a MOVING violation such as speeding). None paid the ticket for those I know that received tickets despite complaints. Even if they did, who is going to pay for the increase in auto insurance? Better check these items yourself anytime you are getting into an unfamiliar vehicle.
One thing that may help in traveling is having a cell phone. Even the best maintained vehicles, no matter how new, can have a break down.