Ashke, In my experience "try-outs" are generally a waste of time, but often a token warm fuzzy feeling for all those involved. The head coach of a group essentially knows who they are going to put into Div 1, Div 2 etc before the tryouts so why pretend. So really the only true "tryouts" where talent is unknown are probably the U10's or U12's.
Closed vs Open tryouts. Mmmm. The theme that comes through is that someone seems threatened by having parents watch "tryouts" for fear of criticism? May I suggest that perhaps a more subtle approach is needed. While the kids are "trying out", the parents could be diverted to a breakout room and provided with the necessary information for District. Parents Handbook, how District works, Q&A, or even scoretable training (No friggin' excuses from lazy parents then!) Then, if all things are timed to perfection, the parents miraculously appear just as the "tryout" is completed. That way you have your closed tryout without parents watching, and the parents are kept up to date with the latest information. You can't say that couldn't be made to work.
Alternatively, at the start of the tryout you could get parents to stand in the middle of a pentagram, chant their name three times, clap to make them disappear, then repeat the process when you want them to re-appear.