I think people continue to breed their "oversize" horses because those horses are likely to produce foals that are small enough to be registered--"legally" registered. That is why you hear people saying that if a horse constantly produces oversize foals that horse will be automatically removed from the breeding pool (or however it was worded above). It's one thing to breed a 39" mare that always produces foals 36" and under; there would be no point at all in continuing to breed that 39" mare if all of her foals end up at 40". Yes, an occasional oversize foal might not eliminate her, but for most breeders it would eliminate her if she produced oversize foal after oversize foal.
Would I turn in my papers if a horse went over 38"? Probably not, just as I don't bother to send in the papers on any of my deceased horses either. I'm not using those for any nefarious purpose, and if I kept the papers on a 39" horse I wouldn't be using those for anything dishonest either. I wouldn't be trying to show the horse, and I wouldn't be bothering to breed the horse, and I wouldn't be trying to sell the horse--with or without papers. If the horse is just here looking pretty standing in my pasture, or pulling a cart around the countryside, and the papers just sit in the drawer for the next 40 years, there's no harm done.
My view on why showing an oversize horse is dishonest--because you are outright lying about the size of your horse. You are saying he is 38" or under and showing him in the 36" to 38" class, when in reality he is 40". I do have a problem with that. If however you are breeding that 40" horse and he's producing 35" offspring, I guess I don't really much care (And remember, I'm not one that is doing this, so don't jump all over me for my "dishonesty"!)--those offspring are legal for registration purposes and for show purposes. Sure, I suppose that someone lied at the time the horse got his permanent papers, or maybe he just grew some more after he got his permanent papers, but if he is routinely producing "legal" size offspring, I just don't have a problem with that. I personally don't think a horse should lose it's papers if it goes over the size limit; I think it should still be considered breeding stock, even though it cannot be shown.
Do you know how many times someone has said to me "if you have a horse that goes oversize, don't turn in the papers, you can still breed it". I recently bought a pony and commented to someone that she is too big to qualify for AMHR. That someone immediately said OH! Whatever you do, don't turn in those R papers!! I pointed out this pony does not have R papers, so there are none to be turned in! But yeah, there are a lot of people that don't believe in turning in papers. Some of these people didn't always think that way--I know very well that some of them have turned in papers in the past, but have since changed their way of thinking. I guess they've seen so many other people breeding (and showing) oversize horses that they have decided that's the way to go.