I had a string of five flats in just a few weeks (after none ever for a year or so before that) a couple of summers ago. Turns out that when my husband put rim tape in my new wheels, it wasn't quite wide enough, so I guess some spokage was coming through. Once he put on wider tape, problem solved. I don't know if that's the problem here but glad you're going to change it out, just in case.
And yes, the more he weighs, the more tire pressure he needs to run to prevent pinch flats. But since you say that most of the punctures are on the rim side of the tube, I suspect something with the rim. Inspect it closely!
Good luck,
Emily