It's distinct from club to club whether the coaches have the right to choose players they want in their team. I don't know how Manchester United handles this, but I think Amorim has something to say there. One of the biggest problems for Manchester United now is that they did not qualify for the Champions League and that makes a big difference for most of the best players when they have to decide where they want to go next. It is not happening very often that some of the best players are willing to play a season without Champions League or Europa League. Additionally they don't know whether it is only for one season. If it doesn't go very well in the next season, it could be years for those players without the Champions League.
Starting with whether the manager can have ability to hand pick players, yes it is usually what happens in some clubs and with some coaches they are allow to hand pick their own players themselves but for most of the clubs I know they do not allow managers to hand pick player by themselves but rather ask the manager for the positions he needs reinforcements and then the club’s scout will now go and scout the market with the profile of the player they need and then Pick out the names to bring back to the club and then the manager is let known of the player and the club and the manager comes to a common ground base on the transfer fee, wages and other details.
Yes clubs like Manchester United do not actually have to qualify for the Champions league to attract some players to the club because they already have a name that actually lures them this players, most of the time it’s some player’s childhood dream to play one day for the club and even with champions league this opportunity isn’t turn down. Some of the players also have this believe that they can help bring the club back to glory and with the project of the manager presented to them they simply sign for the club
No it is the opposite of what you just said. I have no doubt that players would decide against Manchester United when they are not qualified for the Champions League. You can't think that missing the Champions League has no impact on player decisions because it definitely does. Try to convince one of the stars to go there when they can't play Champions League. You can perhaps do it with one or two players because you overpay them and they accept that deficit in exchange for money, but in a normal situation they would not do it. Players like Haaland would have never joined Manchester City if they didn't play in the Champions League.