My wife used to work for the HOA in Eastlake. She told me that exact same story about 5 years ago. She said the same thing. But because the neighbor could see it the HOA lawyer said it had to be taken down. And it was.
On top of that you are at the mercy of the “board”. Because not enough people show up for elections board members are allowed to appoint people to vote for entire area’s of the HOA. So good luck with a fair election. Also board members plans are rarely turned down by the board.
After her stories I refused to buy in an HOA neighborhood.
So the only real way to win is to run for office.
Good Luck
It really sucks that your neighbors hate the sound of children having fun.