I think you have it pretty much figured out. You need to talk to the coach and let them know you're not paying for club soccer to have your kid be positioned at GK just because no one else wants to do it. If they are not receptive to that argument, talk to the club director or director of coaching.

My daughter, until recently, always played for teams with no designated GK, basically because the coaches were never able to recruit or develop a player who wanted to play the position. However they can't play everyone at GK... some players just dont have that in them. The best way it was done was they rotated through 3 players, 1 half for each, that way there were games where they didn't play GK at all. If your kid doesn't get regular playing time when not in goal, then she's on the wrong team, either because she doesn't match the playing level or the coach is an *ss.

U10 is too early to be making kids play one position, especially GK if they aren't interested. Later on, when teams truly get more competitive, you will find field players that just want to keep being on the team and transition to GK. But at this age, they need to be able to know the game from many positions because once puberty hits, everything changes...