Right now we're approaching about 17 million out of 21 million possible bitcoins. When we reach the max supply,
More than 110 years from now? After we are all dead? Why are you worrying about that?
what incentive would miners have to continue mining and verifying transactions?
Fees.
Sure they can still collect fees,
Exactly.
but why not mine newer and more profitable coins where they have the chance to collect both fees and new coins?
Would you rather mine 1 coin worth of fees at $100,000 value per coin, or 100 coins of reward at $10 value per coin?

What's stopping the entire blockchain from grinding to a halt, prohibiting users from conducting transactions, and thereby making bitcoin worthless?
Demand. Usability. Profit.