I think every situation is different. A lot has to do with how and why the team was put together. In most cases the player picked up is a pitcher or two. This, in fact, could be so the coach doesn't have to abuse his pitchers. It could be because the coach can pick up a big prospect that will draw lots of attention from scouts and college recruiters. This giving others on the team much more exposure.
If the team is formed to simply find out who will pay to be on the team and those players are told they will play all the time, then that is what should happen. Those kind of teams should not get involved in highly competitive week long tournaments. They simply don't have a big enough roster.
in most any situation where a pick up player is used, someone needs to sit. In most, not all, cases this would be the weakest player. Baseball at the very highest level is not a pay and you play thing. The rosters are large in order to compete and someone has to be on the bench. The team plays a lot of games. Those on the bench are talented. They play when the coach decides they will play. When they get that chance they need to take advantage of it.
So I think there are different ways to look at picking up players. It all depends on what the expectations are from day 1. But what do you if you have a chance of picking up a potential first round draft pick? What happens if you leave it up to the team? What if 14 of the 15 players on the team vote in favor of picking up this player. Should the one voting against it get his way because he paid to play? You're the coach and understand that several of your players will have a great opportunity because that kid is going to pack in the scouts and recruiters. Or do you think about who paid to play?
Guess I can see a good reason to go either way. Then again I would never tell anyone they play all the time if they have paid. And I would never have allowed my kids to sign up for a team like that. Playing time is one thing and everyone should play, but coaches sometimes do what is best for the group, rather than the individual.
My youngest son when he was 13 went to a National Tournament with a team. I paid the same as everyone else. He ended up playing one inning. At first I was pissed, but they went to compete and he was the least talented player on the team. And he told me he had a blast. Did it harm him? 9 years later he was making his debut in the Major Leagues. Guess it must not have hurt his development all that much. But who knows, maybe he would have been better had they let him play every game at that 13u tournament.
I will admit, if my son were the odd man out because of the team picking up a ringer, I would be mad. But I wouldn't expect everyone else to be mad about it. And I wouldn't want him on a team that decided playing time and positions based on nothing other than paying a fee. I know it happens though and in that case no player should be replaced.