I was given free reign as a kid at the cottage (early/mid 70s...I was about 8 years old) and while being responsible seemed hard-wired in my case, my buddy Mike was a little more...carefree should we say. I do remember an OPP cruiser pulling up beside us one day as we strolled down a dirt road. The officer took our names, told us to be careful, and not to walk with our guns loaded. (which we never did anyway) Then told us to have a nice day. Imagine that ever happening today? lol I suspect some cottager saw us...thought our Chinese break-barrel rifles were AK-47s.
To the question~yes, depends on the kid I think. However, "critter" should be more clearly defined. In Ontario, there is a rabbit season and you need to be a licensed hunter with a small game tag to go after them. Even groundhogs here require a small game license whereas in places like Alberta and the lowly gopher~I don't think any license is required. If the animal is a pest animal around the farm (ie, pigeons/gophers/rats/mice/crows/magpies etc.) then I doubt a kid being careful with an airgun would ever be a problem.
I think .177 is slightly less useful than a .22 cal. air rifle, but, effective enough if the critter is small/close and the shot is good.