Do they make child seats for saddles like they do for bikes?
They do actually. I've seen little bumper seats that do fit on like a saddle bag to give a little security of seat, and a little handle to hang on to. But you wouldn't really want to strap them in - there's times when the best thing that can happen is for the kid to get bumped off early in the event.
I've seen women ride with babies in those front-papoose things. Not sure it's a good idea, but I can understand the desire to get out and do things.