It is completely fair for the new manager to decide who is on their coaching staff. I am sure they will give consideration to the existing coaching staff as part of the transition. It usually happens that way.
As for Stairs, we still have no evidence that he was any good at the job. It is not as if many of their existing players made significant progress. The big improvements to the team have come from new players on the minor leagues. Way too much is made of a coach's supposed philosophy. To some degree most coaches preach 95% of the same stuff and I am guessing Stairs was no exception. The big test is whether the reach players and if they improve.
Looking at the existing players it is hard to argue that Joseph, Franco, Galvis, Hernandez, Rupp, and Herrera improved at all. Some of this is the players themselves, but there is no evidence the coaching made much of a difference. Altherr might be the one holdover that took a step forward. Williams, Hoskins, Alfaro, and Crawford are much more the product of the minor league coaching staff.