At the moment, Everett is a "C" TE. Sure, he has lots of athleticism, and "potential". But so far he's not lived up to it. Being regularly dinged up doesn't help; but that's his problem to overcome. Flashes won't get you a fat FA contract; but the Rams probably aren't even willing to pay him as much as $3mil per year, and I would imagine he can get more than that.
Something would have to change big time for the Rams to keep him after 2020. They didn't draft this new TE to be a 3rd string player for more than his rookie year. They've committed to Higbee as their centerpiece TE. He is living up to it, and I imagine this coming year he could become the Gronk of the Rams.