Handing in your notice early like Powell and Price allows both clubs time to look around and get the best available coach for them. It also means the respective coaches can decide who they wish to retain in their new clubs. Otherwise from ? May 1st players who are on the end of their contracts will look elsewhere, and may find much better offers as well. So I think that all seems to make sense.
Should they also say who they would like to be recruited as well as retained at their new club ? I don't see why not, the new coach will wan't players that suit his style of play as well as filling gaps that he sees in the current squad. Should they try to recruit from their current clubs squad ? I would have thought that they would talk openly with their CEO and refrain from encouraging players to migrate with them. However, if they do want to take a player then the best approach would be to talk to their new CEO and say who they want, then tell their current CEO, but avoid discussions with individual players. I am not sure you can do much more than that, and at the end of the day it will all boil down to the level of trust and respect between the departing coach and his old club.