This might happen. Or they will be shown on local circuits where there are mostly zero drug regulations (California is an exception since the state does drug testing; there may be others). Or end up in a non-show program, or as a practice horse for someone who shows. Or become a lesson horse. Or get sent to an auction, possibly ending up in one of the above scenarios. Or get nerved. Or get retired or semi-retired. Or get euthanized (I’m distinguishing euthanasia from slaughter, with the latter occurring in a slaughterhouse).
If the OP’s hypothesis is that the horses that are shown lame or medicated are owned or controlled by people who don’t care, it would follow that that these same people would be likely to dump the horse.