A new manager is not going to need a month on the training ground to figure out that Moussa Sissoko is a donkey who can't pass the ball or that Eric Dier is no better than a traffic cone. Yes, having more information via training sessions and personal meetings is a better thing than not, but it's not something we need in every case.
You're also only talking about outgoings when one of the key things a new manager will do is bring in new players. A new manager (and hopefully a new DOF) will be spending a lot of time figuring out who they want to bring in and who they can bring in. This requires a ton of analysis, negotiation, etc. and it's stuff that can all be happening right now.
Guess what, other clubs don't take holiday while the Euros are going on. Every day that passes without a manager means the new person has less time to be ready for the season.