TBH, if they are closing the grounds because it has rained, they are probably covering their arse! An injury to children (or anyone) whilst playing sports on a wet oval of the council's is just not worth the insurance or the hassle... better to postpone or cancel the game.
The Insurance for players would be covered in their Registration Fees through the sporting body they play for ie soccer, AFL, Rugby League etc and is not the responsibility of Council usually.
What you'll probably find with the ground closures is that it isn't the Council that are closing them, but the sporting club themselves. A lot of Councils lease the Clubs the grounds etc but you'll find the great majority of maintaining the playing surface is done by the Club (and usually by volunteers) and in this financial climate the Clubs simply can't afford to repair the field every time it gets torn up by letting the kids play on it when it's too wet. It can take weeks to get the grass to regrow over torn up areas in summer let alone winter when everything grows a lot slower.
I've been on both sides of the fence with kid's sport, both as a mum and Executive Committee member, and I know how much work goes into maintaining a safe playing surface....it isn't as simple as mowing and watering LOL