With shimano at least only the (guide) pulley closest to the cassette has deliberate side to side play. The lower (tension) pulley doesn't. That's why higher end shimano mechs use a cartridge bearing lower pulley and a ceramic bushing in the upper pulley. It's hard to engineer side to side play into a cartridge bearing pulley.
I tried Forté style pulleys once and never again. You can either fix them in position so they turn on the bearing and have poorer shifting. Or you can fit them loosely and inevitably they turn on the bushing instead of the bearing.