And is the cooler in cars big enough to have noticeable towing capacity
Yes, it is. Do you realize that manufacturers publish a maximum towing capacity as part of their specifications for every vehicle? This is publicly available information, right there on the internet. It's not a secret. The required surface area for the cooler is designed right in by the manufacturer for the transmission to work for the vehicle's application. This not a case of something "extra" being added. It's just how cars with automatic transmissions are built to begin with.
The published towing capacity for most vehicles that are available in both automatic and stick are exactly the same. Would you care to guess why that is? You could have figured it out for yourself if you would bother to actually do some extremely minimal internet research instead of continuing to shoot your mouth off on whatever this ill-informed little crusade of yours is.
Your initial claim is false. End of discussion. Just stop. You're making a fool of yourself.