If you live in an HOA with rules such as the kind of driveway you can have, I'd be surprised if they don't have strict rules about keeping ones pets in your own yard. Are there any ways to enforce via them?
I agree that the issue is their being in your yard, beyond the pooping. Even the most lovable dog could react badly around the sudden movements of a child, and I would not want someone else's dogs unpredictably in the same place where my kids were playing. I think playing this angle could be to your benefit - it would be terrible if the dog had to be put down because it was in your yard around your kids and bit one of them. (To be clear, I can imagine with some dogs not being too worried about this, at least if I were supervising the play/interaction, but this could be something to express concern about - who is liable and what are the consequences of their dog biting one of your kids - that might resonate more with them than the supposed little inconvenience of some poop.)
I agree that the issue is their being in your yard, beyond the pooping. Even the most lovable dog could react badly around the sudden movements of a child, and I would not want someone else's dogs unpredictably in the same place where my kids were playing. I think playing this angle could be to your benefit - it would be terrible if the dog had to be put down because it was in your yard around your kids and bit one of them. (To be clear, I can imagine with some dogs not being too worried about this, at least if I were supervising the play/interaction, but this could be something to express concern about - who is liable and what are the consequences of their dog biting one of your kids - that might resonate more with them than the supposed little inconvenience of some poop.)