quickness

You mean foot and calf strengthening work so you don’t snap your ankles?

Your feet and calves act as shock absorbers and a link to transfer the power from your hips and legs to the ground.
If they are weak you will not being able to absorb the forces well, and can cause shin splints and other problems. And you won’t be able to stay on the balls of your feet and be springy and reactive.

Rate type work is great for being able to be light on your feet. All kinds of simple low intensity hops on the balls of your feet concentrating on relaxation and maximal rep speed for time - ie backwards and forwards, up and off a low step, quadrant etc

But for actual strength and force absorbtion - then things like backward semi-stiff legged altitude landings for the velocity side (use low heights), and usual calf work (ISO, oscillatory isometric etc) for the force side.
These are some of the things I am doing

Please explain how the natural movement of a false step is “bad”?

It’s a wasted movement. You can tell when you see unathletic people playing certain sports. They make alot of unnecessary movements.