I've tried 70-200 f2.8 IS II + adapter on eos-m Vs native 22mm. AF speed is not the same. Why? I can't tell you.
Neuro confirmed his adapter + L lenses is faster then native 22mm. Maybe, other members can give their feedbacks.
Hardly a valid comparison though as it is two different lenses. It doesn't demonstrate that the adaptor makes a difference....
The fastest focusing lens I have used by far on my M is the EF-S 55-250mm IS STM - via the adaptor - it is faster focusing than any native EF-M lens hence demonstrating the adaptor has no impact I can see. My belief is because it is able to house a bigger STM motor than the EF-M lenses, and the M is optimised for STM lenses. It is faster than my EF 70-200 f2.8 IS II. Nothing to do with the adaptor AFAICS.
So, I can't see any reason that a new adaptor is needed for the M