I think the F1 teams would all switch to racing versions of those transmissions the second they would be allowed to do so.
The efficiency gains wouldn't even be important in comparison (until you start bringing significantly less fuel than your opponents), but just the reduction in weight and size (important for aero considerations) would be worth it. Also, the power gains from always running the ICE (and its turbo) at the perfect sweet spot in the power curve would be a giant advantage in racing.
Yeah, but a huge part of F1 is the sound. It's iconic in the true sense of the word. I can totally understand them wanting to protect it as part of the brand. Enthusiasts might be into other types of motor racing, but F1 sounds fast. Everyone understands that.
Correct, it has been done and got shutdown immediately.
https://en.wikipedia.org/wiki/Williams_FW15C