It's been a couple decades now, but I rode my toddlers around on the back of comfort bikes (seat on top of rear rack), big tires, relatively heavy frames, upright position. I always felt stable with that setup, and while handling was different than without the seat, the bike was just as controllable with very little getting used to, a few minutes on a safe path should do it.
I agree that I would have found drop bars problematic if for no other reason, leaning too far puts your butt pretty much in the kid's face.
Whatever you get, I definitely recommend relatively big tires, a jarring ride is not your friend.