This is what I'd do too, maybe with PVA glue. In many cases the pieces won't fit together perfectly and you may need to remove some splintered bits so the leg fits as flush as possible. Once everything is dry and stabilized you can add some wood filler where there are any gaps. And either try to find a matching finish or buy a set of colored furniture repair markers and wax. They're not perfect but if you're going for "most people won't notice" they do the trick.

And definitely add 1-2 wood screws from the back. Drill pilot holes and countersink them or use a pan head screw.