I am certain that the car would not break if you towed it for a short period with it on and in drive.
However there are a lot of ways that you could break it by towing. Mostly:
- If the car is off and you tow it, you could fry some of the electronics
- If you towed it for too long or at too high speed it could overheat and damage the motor or other electronics
The section in the manual is addressing the "typical" cases where a car would be towed: broken or behind an RV. Both of these cases would not have a driver and the car would be off which would absolutely cause damage.
They could allow tow charging but that means they'd have to invest in the testing and take on liability for damage which likely wasn't worth the benefit of allowing a couple of owners to tow charge in an emergency.
Another semi-related example of manufacturers "banning" something for no reason is towing. Tons of vehicles are sold in the US that say "not rated for towing" or "do not tow" yet the exact same vehicle in Europe is rated to tow 1000-1500lbs. The only difference is that Europeans expect cars to tow so manufacturers go through the trouble of testing.
Edit: don't take my word for it though,
MotorTrend asked Ford, Rivian, and others