No, this is why Transport Mode, which keeps the car in neutral, for free rolling, exists, to prevent damaging the motors.
Nope. "Tow/Transport Mode" just releases the e-brake ("park"), puts the car in neutral, and most importantly doesn't re-trigger 'Park' when you get out of the front seat. Rolling the car in 'park' is just resisting against the e-brakes, not the motors. That's it. There's no "Park" in the motors themselves. You can't damage the motors by pushing them in any mode. Where did you get this info? Do you know how electric motors work? They are not like ICE transmissions with parking pawls (as I said before). I think you seem stuck on thinking that's how EV motors work. They don't.
I can roll the car in drive, neutral, or reverse and not damage the motors. I can't roll it in park because the e-brakes are applied, and that would take a lot of force to overcome.
Have you not seen the people posting above about going through the car wash in "Drive" and being pushed by the rollers? According to you, that would damage the motors. Doesn't happen.
My car wash place immediately knew they had to turn on Transport Mode.
You don't have to use tow/transport mode. You can, but it's not necessary, but you have to stay in the front seat. I drive through car washes all the time in neutral. My motors aren't damaged. I'm going to try 'drive' next time, though.
The car in park will keep the e-brake applied and be dragged, causing damage.
You love this word 'damage'. If the car is in Park and the e-brake applied, in a car-wash scenario, the roller will simply lift up the rear wheel and slide underneath. If the track is sufficiently wet and slippery on all four wheels, it's possible the car would slide along even in park, although that's not likely and I'm not going to test that. But I've seen the car 'hop' the roller while in park numerous times. That's how it's designed. No 'damage'.
Has anyone in the history of Tesla and car-washes ever had a motor damaged? Nope. If Tesla motors can be damaged by simply going through a car wash the wrong way, they did a phenomenally bad job at designing them.