For karate, important uses of the gluteus muscles include:
explosively closing distance, particularly when the back leg is bent and the knee
is turned forward, so you can think of the back leg contributing
something akin to a sprinter pushing over the starting block,
delivering a strong, explosive jab or reverse punch (the glutes will
tend to contract on ...
From a non-karate perspective: the direct connection is better defined by force transfer than stance geometry. Imagine someone pushing back into your punching arm. If you maintain your shape and balance as this force is applied, either the force will get transferred instantaneously into the ground or you will slide. When you remain in place (don't slide), ...