I'm not even making an argument. I just genuinely want to know if an incapacitated creature can unstuck itself.
Codex for Incapacitated gives a laundry list of things the creature can't do, but doesn't list escape rolls. Playing as written I'd guess that a Pit Trap can wear off even if the creature is incapacitated, because an escape roll isn't listed as an action: only quick, full, and free actions are actions... as far as I know.
Rolling for burn isn't an action either, and incapacitated creatures burn just fine, but rolling for stuck and cripple happens during the creature's action, which might be different?
In the light of dawn, I'm now leaning toward saying that the creature does roll, because it's not described as an action.