I am a long time cyclist and have degrees in physical therapy and biomechanics. In my experience, I have never seen a cycling shoe with significant ankle support. Some winter shoe models cover the ankle, but don’t really provide support. The motion of the ankle varies from person to person and no one particular strategy has been shown to be the best. This has also been called the “ankling” strategy. It refers to the up and down motion of the forefoot in relation to the heel while pedaling. Many riders endorse moving the cleat attachment under the center of the foot to improve power transfer from the lower leg and decrease the ankle force required. This has not been backed up in any studies to my knowledge, though.
The general theory is that the muscles that cross the ankle are much weaker than the upper leg / hip and knee extensor muscles. As the ankle muscles fatigue under hard efforts, it’s reasonable to assume that ankle motion would dissipate energy through eccentric lower leg muscle contractions - therefore an ankle brace could aid in power transfer from the upper leg. This is very similar to why cyclists use tight, stiff soled shoes - any intrinsic foot movement would simply waste energy.
I don’t know if it would be uncomfortable or not to have an ankle brace on the shoe - there’s never been one to try. I imagine that if it did help power output at all, the pros would be using it for time trials, mainly - where comfort really isn’t a significant consideration.