Can you clarify by what you mean as a "team first" reaction? If someone such as Mourinho comes in when the sun shines tomorrow (depending where you are!), players will be scared and have to change their tune. Every player will look at for themselves, unsure at what to expect because only Mata has worked with Mourinho and he was sold by Jose.
United are still in the FA Cup, still have to play in Europa League with 4th still (mind boggling that this is true) in sight. A new manager, assuming it's Mourinho, will probably relieve Giggs and a few others. So you're inserting an entirely new coaching staff, with zero insight. Nothing. You're coming into the environment blind.
Beginning Feb. 18, United have 4 matches in ten days, not including a potential FA Cup replay at OT. Then in March, United could have up to five matches in 18 days if they qualify for the sixth round of the FA Cup.
The coaching staff will be changed, but with everything in football and life, timing is everything.