Need a bike I won’t fall off

I was a keen road cyclist for over 40 years, then switched to trail running after a serious concussion five years ago (my second – the effect is cumulative, so I don’t want another). Now I’ve sprained my ankle and probably won’t be able to run for a couple of months. Physio friend recommends cycling to maintain fitness, so I’m thinking of getting another bike. I want something a lot more stable and forgiving than an ordinary road bike. Needs to be cheap too. Any ideas? It’s for on-road rides of an hour or two and I won’t be doing serious hills on it. Thinking of maybe a rigid, single-speed mountain bike with slicks? Ideas for which make/model to buy would be good.