this phrase is used in sailing, skiing, motorcycling, showjumping, skydiving, tennis, classical musicianship, downhill mountain biking, and lately I hear, mma. I have some doubt that it originated with door kickers.
I suspect it does originate in the military, though, and probably reloading guns and cannons while under fire. Might go back further than that to bows and arrows.
Looks like a related term "make haste slowly" dates back to Roman times and was applied to military commanders: https://en.wikipedia.org/wiki/Festina_lente