Yeah, the use of 3 doors, and the bit where one is opened just kinda hides what’s happening. Which is that you’re actually choosing between one random door, or ALL the other doors.
The Eureka moment for me was similar, but slightly different. Consider a variant game with N doors. You pick a door, then Monty offers you two switch your payout to "the best single result from the remaining N-1 doors". Clearly, you would swap.
Now set N to 3, and consider that, by opening a door without the prize, that's exactly what Monty's offering you.