I think most players are a bit more intelligent than just weighing up how big a "name" their coach is. Who would you listen to - Lage, or Lampard? Wenger or Shearer? Mourinho or Ince?
Obviously if the coaching is crap, that's a different topic entirely.
Hold on.
Ince - great player **** manager
Mourinho - no player, great manager
Wenger - **** player, great manager
And more
If it was just so simple Mourinho & Wenger would never have been coaches and Paul Ince would be a top coach