Several reasons for it:

1. Minimal wear and tear on the body. Unlike most positions, you don't take regular hits and you are only on the field a few times a game usually. Only real risk is someone diving at your legs and messing up your knee, but punt block teams are very careful to avoid that since getting a penalty in that situation is equivalent to a turnover.
2. Longevity. If you're good, you can easily play into your mid to late 30's. Kickers and punters have the longest average career of any position.
3. Low in-game pressure. Unlike kickers, a punter doesn't make or break the game. Rarely do you point to a game and say "If only our punter pinned them deep like he was supposed to." Even if a punter shanks one, which does happen (although not super common in the NFL), the defense can still go out there and make up for it. But for a kicker, a win or a loss often comes down to them doing their job in high pressure situations.
4. Low competitive offseason pressure. I mean this to say, if you are an established punter, your team's front office is not going to worry about bringing in competition to push you. They aren't investing draft capital to bring in dudes to compete with you for your job. Punters are rarely drafted. They are focused on upgrading literally every other position ahead of upgrading a punter who is already doing an adequate job. Obviously breaking through to secure one of those starting punter jobs is extremely difficult due to that longevity and low attrition factor, but if you can secure a job and do it well, you are sitting in a terrific spot.

If I had a son that had good size and was playing soccer or something at a young age, I would try and steer him toward punting. Most kids who play football growing up aren't dreaming of being a punter, so competition at that age isn't crazy high. Someone who specializes in it starting at an early age has a good chance of going on to play in college and beyond IMO, much better than than playing any other position if you aren't an athletic freak.