Yep, possible. Whether the person or the spacecraft is accelerating (rotationally in this case), the perceived force between the two is the same. See this for some video proof (I am stealing the second link from the comments - thanks @JCRM). Notice that to begin the runners pull themselves into a crouch and push off in the direction they want to run.
How Does it Compare?
You touch on this in your question, but ISS astronauts exercise for a chunk of every day and they still have issues with bone density loss and muscle atrophy when they return. So replacing one exercise for another wouldn't alleviate these issues unless astronauts exercise longer and can isolate all muscle groups. You are definitely right that running in a circle is simpler, but it would be hard to isolate and exercise a lot of muscles in the upper body this way, and you would still need some of the exercise machines astronauts use on the ISS.
Here's where there is a notable difference in your idea: Things like bone density loss and muscle atrophy are helped by physical exercise, but other things like changing in the shape of the eye's lens are not helped by this. These effects are due to the "microgravity" environment and would be only alleviated by living under 1g (or nearer to 1g). So it's possible that spending all or part of their exercise time effectively under actual acceleration would help lessen these effects.
I think at the end of the day, the running is much simpler than rotating the spacecraft/space station. However, you are only simulating gravity for as long as you are running. Spinning so as to effectively simulate 1g (or a decent fraction of that) allows astronauts to live in an environment much more similar to Earth, and alleviates all of the negative effects of "microgravity" with little individual effort. Unless we figure out how to deal with the negative effects long-term, constantly rotating habitats seems the most likely solution by avoiding the problems in the first place.