Depending on where you live, it might be nice to see a lot more of what’s around you on a bike. It’s similar to OP’s idea of getting lost in the flow of a competitive sport. You get lost looking at all the places that pass you by. You can take random routes each time and learn the city around you pretty well. It doesn’t hurt to have your headphones in and listen to an audiobook while you’re doing this either.
I never really got into running because it’s so much slower than biking. You will see a fraction of what you see on a bike, so I think it probably gets boring quicker. Plus it puts more strain on your joints.