I'm glad you decided to raise this problem after many years of organizing Faf tournaments.
In order to divide the participants into a large number of rating groups, you need to have its own prizes for each group. In addition, this option does not exclude manipulation with the rating in order to get into a lighter group.
If you tell people the rating limits after the fact, then next time no one wants to participate. Because many will want to play in the tournament, will cancel their plans. And will be very upset when they are not allowed
It is best to run tournaments with captains. But even with the idea of captains, people will still manipulate ratings. Because the selection order will depend on the ratings of the captains and their teammates. The only option left is to give the teammates at random or just the player with the highest rated player with the lowest. But this excludes the possibility of playing with friends, limits the format of the tournament and requires a large number of top players as captains.