There are two things we can't make another human do no matter how much we want to -- sleep and use the potty.
Getting up is so common with the transition from crib to bed. Yes, he still needs some extra sleep,but he may be ready to give up the nap most of the time. Not what any parent wants to hear, but it could be true. My daughter pretty much gave up napping -- other than in the car -- around 2. I did take her out for a drive most days after lunch and she got that nap she needed, though it was shorter than in the crib-napping days. That lasted a while, then we went to quiet time.
If cruising naps aren't for you, go with what another poster mentioned and give him "quiet time." You can still start out with him on the bed, read briefly to him (I'd give up the "reading until he can hardly keep his eyes open") and say, "It's your special quiet time now" and leave promptly. If it is a routine, and you continue your very good idea of not interactiing with him when he gets up to find you, it'll work eventually -- to teach him to be OK alone in the room, not necessarily to sleep. You could try putting a gate on his room doorway, so he can have the door open enough to know you're there but not to get out and find you.
In other words...keeping him in bed may no longer be an option. Some days he may sleep on the bed, others he may fall asleep on the floor with his toys, other days he'll cry for you because he realizes he's not getting to play his get-up game. When that happens I'd go near the gate, give a quick reassurance that you're there but not pick him up or engage him much, and leave again. And he'll be cranky some because of the lack of naps. But short of driving him around or exhaustiing yourself trying to force a nap he just won't have, quiet time sounds like a next step.