That’s also not true. State law in my state and I’m sure others requires them to confirm you have proof of insurance before you take possession.
Doesn’t have to be on the Tesla as your existing policy will provide grace coverage of a new purchase for at least a week or two.
Proof of liability coverage, yes. Proof of collision/comprehensive on the vehicle you're purchasing? Nope. That's only if the lender requires them to verify that before releasing funds.
And they verify that because the grace period you mentioned only happens if you already have collision/comp coverage on another vehicle. If you own your vehicle outright and have liability only, you don't automatically get collision/comp on the new vehicle. The grace period effectively provides the "best" level of coverage that you have on any vehicle on your policy to the new vehicle.