If your box doesn't smell, I'm sure it's fine! As long as you haven't found any "presents", there shouldn't be a problem letting your children use the box. Just be sure to dig through it very well.
As for the people making digs about letting cats outside, you're not being very thoughtful. While I don't agree making this poor cat stay outside all day, and NO ONE knows where she belongs, (and in this case it really does sound like a neglectful owner so animal control really needs to be called or you can set up a human trap and then take her to the pound yourself) that doesn't mean all cats that are outside are horrible dirty little creatures. All of my cats come and go as they please, they stay pretty close, they're "clean" and up to date on all of their shots, everyone along our street has put out cat deterrents, etc. Everyone knows what animal belongs to who on our street, so if there's a problem with a pet, we hear about it really fast. I have never caught my cat going outside, although I'm sure they have. But there's enough "wilderness" surrounding us, that it's never been an issue. Even in the garden and big plant containers, they don't go in it. I just have lots of orange peels in the areas that I don't want them and they're good girls.
http://www.cat-repellant.info/html/diy-cat-deterrents.htm