And that park is mostly surrounded by homes
So?
The mouse could very well be eaten by a natural predator, or live the rest of its life under a log eating insects and algae. It could very well never enter any of these houses. And there'd be other mice in the area, most likely, hardly matters. We have a natural reserve nearby with mice... and guess what, we haven't had them for over a decade.
Oh yeah, let's all go and release our house mice in the park!
Not a good plan. Now you'll have much more mouse pee and poop than usual in the park for all the kids to touch while they are playing.
With that mentality you'd might as well put your child in a bubble and never let them go outside, because they might catch something from playing in the soil.
Whether we like it or not, mice will be in the area regardless so it's not going to matter one iota if you release it in the park. There will very likely be other mice there, not to mention poop from other animals - feral cats, possums, birds, insects, other rodents, etc
Where I live, rodent-carried leptospirosis is an issue. Dogs and humans can catch it. We would not want to encourage rodents to be where dogs and kids play.
Then do not let your children play in soil with any open cuts (because that's how it is usually transmitted). Do you have any feral cats in your area? They are known carriers too. Dogs are also known carriers, in fact 25-30% of dogs and cats are carriers of this disease. Your dog could very well be a carrier too.
People worry about disease when mice carry no more disease than any other wild animal. The major issue with mice is their breeding rate if they make a nest in a home, and their habit of gnawing of wires. To get a disease from one is rare, or for one to be so absolutely careless as to let them run on the kitchen counter for lengthy periods of time.
The whole point of releasing them is for them not to be a problem any more; you're returning to how it was before that mouse was in your home so where ever it goes, is completely irrelevant. People will kill them because they're too lazy releasing them, it's more convenient for them. I feel that killing is a last resort measure, and I don't agree with it unless you have an infestation (and that means, dozens of them). But if it's done humanely, I'm OK with that even if I might disagree. The main thrust of my criticism is the use of glue traps, and the OP suggesting that that'd let their husband throw a live animal still stuck on the glue trap into the rubbish. Now
that is completely unnecessary and beyond cruel.