I instagram photos from when I run, but rarely of myself (unless someone else gets a photo of me - sometimes a runner makes a good subject matter in a photo if it enhances the rest of the scenery).
Why do I do it?
1) I like photography, even if it's from a bad phone I enjoy trying to capture a good shot from runs. The bad camera phone makes it more challenging, but I love the challenge. This applied to my running before instagram as well.
2) It gets me out running more. When I was young, chasing PRs and wins in track is what got me out to enjoy running. I'm never going to PR again or accomplish anything in running I'd be super proud about at this point. I do love trying to capture good nature photos, and running to places not otherwise accessible is a great way to do it. It's fun and keeps me in shape. With 3 kids etc, its tougher to use the excuse to just go out at peak light hours just to photograph, but if I'm "going for an 8 miler" its easy to get out. Also, I'm always wanting to improve on my fitness, not to win local 5k or run a faster but slow hobby jogger time, but because it would allow me to cover more ground in an hour to increase my chances of a great photo. This reason was also true prior to instagram.
3) With instagram, it allows me to publish content I'd be generating otherwise, but keep up with an build connections in the local running and hiking community. I've also met dozens and dozens of other runners and people with similar interests through it. Instagram doesn't replace human social interaction, but I've found it to increase it. Without it there are literally like dozens of local runners I've run with or grabbed beers with that I probably wouldn't have with otherwise. Posting content also lets me receive positive and negative feedback, suggestions for new places to run/shoot, etc etc.
4) In the end it just makes life better, at least for this suburbanite dad, but it really just depends on your goals. If you wouldn't enjoy an increase in social opportunities, only care about running itself and/or getting faster, don't like running carrying a phone in your hand, and would be embarrassed by other running instagrammers, I wouldn't recommend it. (P.S. I agree with much of the points on this thread, but I don't let it deter me from enjoying life or make me care what people would be saying about me)