You can't really compare pull-ups/chins to pushups because they do different things.
As for pull-ups vs chins: Although some say the opposite, most people including myself define a chin as a pull-up with the palms of your hands facing yourself.
Pull-ups will work your upper back more and chins will work your biceps more. But both are excellent exercises for overall upper body strength, and you'd be wise to combine them. I'd give a slight edge to pull-ups if I had to choose because it is in my opinion the one single best exercise for upper back strength, and I think most people will find a strong upper back more useful than strong biceps. Rather hypothetical question however.
Push-ups are fine too, but most people find a push-up so much easier than a pull-up that they will only be useful for muscular endurance rather than explosive strength or "body building". As you know, push-ups primarily train your chest, triceps and front deltoids, so it's a completely different exercise vs pull-ups.
I think you'll still get much of the benefits of push-ups from just doing other stuff such as working the bags, swimming or cross-country skiing. So even though I happen to do push-ups on a regular basis, I'd vote them last in your hypothetical poll.