Speed has a massive effect on turn rate. If you leave aside max rate for moment and look at rate one (3 degrees per second ) the angle of bank required increases with speed. If you a flying at low speeds this will always be 1g or thereabouts. If you keep increasing you reach a point where you will be pulling some serious g hence the normal limit for transport aircraft is 25 degrees angle of bank. If the procedure is designed to cater for high speed operations then it will assume a turn rate of less than rate 1.
In terms of combat manoeuvring speed is still a key factor. The classic modern example is fighter vs helicopter. A helo can sustain a very high turn rate at low speed. It won't be pulling much if any g to sustain this. If turn rate is what you are after then going faster does not help you. There are practical reasons why you don't want to be plodding around the battlefield at low speed in a fighter though so, lie so much else in life it is all about compromise.
When I was learning how to chase another aircraft in a spread formation my instructor called it the egg (small radius at the top, wide at the bottom). If you needed to tighten the turn pull up in the vertical, trade speed for height, at the lower speed you make up the turn you need and then dive back on the speed. You could adjust you position relative to the other aircraft without adjusting the throttle. Not everything was abut pulling more g.