I think by far the main benefit of the quad rule is for airliners: with TCAS you get a lot more time to act.
Um...where to start?
The Quadrantal rule was there long before airways and radar was universal, and was mainly for separating Doves and DC4s.
Airliners nearly all fly on airways nearly all the time. The Quadrantal rule does not apply to airways.
The Quadrantal rule preceded TCAS by a long, long time (let's guess fifty years.)
No, the quadrantal rule is there to reduce the probability of collisions. My own view is that if aircraft flew at a level determined by the last letter of its registration (thus introducing more randomness) it would be more effective, but, as I keep saying, there really is no problem. It ain't bust, let's not fix it.
Will