Two things:
1/ There is not really an onerous task for the coach/manager in sorting out players. Most are familiar with exactly what's out there for the purpose they'll require. And experienced coaches will have 'go-to' agents they trust to sort the details of a deal out. In short: there is no distraction from coaching the team.
2/ Might Brands and Ancelotti differ on types of players required?: they will. How could they not diverge? Brands is someone who hasn't managed, he played to a certain level; Ancelotti was a top player and has managed players for years and knows what he's looking for. There is literally nothing other to do for Brands than to just nod in agreement with the manager who;s here right now on players required. He cant have input, he isn't qualified to gainsay Ancelotti. If he's useful elsewhere in the club (overall football strategy, U23 work, etc) then that's where he should confine his efforts to. IMO if that's all he does here then the club are paying him too much money.
The DoF role here at Everton is a BS position. We've seen waste and chaos after hiring our first 2, and I think that will be underlined soon enough when Brands leaves the club
Do you honestly believe that any club would allow a manager to have "go to" agents that they trust to sort details of a deal out. The vast majority of agents are little more than cut throat mercenaries who are firstly only interested in their own bottom line, secondly the interests of the players that they represent , and a long way last , the interests of managers / clubs. Any club that allowed your suggested methodology of working would be quickly on the way to the poorhouse. Think that you may be a little naive mate.