Your re-framing is actually not quite right either (or at least, not complete). Key point: if the host picks the car, the game essentially re-sets.

Imagine the extreme scenario with 999 goats and 1 car. You select the first door, the host opens 998 doors at random, leaving your selected door and one other. Two scenarios are now possible:

1. There was a car in the 998 doors that got opened. Tough luck, you lose the game automatically now (game resets).

2. There was not a car in the 998 doors. In this scenario, you are still advised to switch.

Edit - I guess I misremembered a simulation I once did about it.