Consider replacing the wood inside if it is suspect of not holding screws tight. Otherwise, fill the holes with some type of filler, I've had good luck with pl-375, but there are plenty of options. This insures the screws tighten properly. Yes, the countersunk screws stay in place, the rub rail then covers them. Rivets can work, but I think #12-14 SS wood screws work better than rivets.