It's a bad situation that you ended up. That is obvious from what you have said. What I question is your assumption that being able to read your LOR's (aka waiving your right to read them or not) would have really helped. This looks more like a cautionary tale of having a backup fourth LOR just in case a situation like this should happen, if possible.
In a case like this, with professors, like students, bad thing happen at the worst times, that they can't get around. It sounds like your professor meant well, but was not able to follow through, and it negatively impacted you. The best situation would have been if you could have had a forth person to look to, in order to give that professor an easy out. But since you didn't and the prof knew you depended on him tried to the best of his ability (which apparently at that point in time not great, and not thinking about the possibility of helping you find someone else to do it) and in the process doing you more disservice then help.
I say this with the fact that he did not give you any indication that he could not give you a good rec. For that you should have a pass. Unfortunately that is not how it generally how it works from what I can tell with ad comms. They tend to see it to some degree the applicants ability to be ontop of everything, including LOR's, which are the most uncontrolable aspect of the application process, as your experience shows. I even at this point have to accept the out come of one application because one of my LOR's didn't finish one of the assignments. I have to accept the fact (in the mind of ad comms) that I was on top enough and started early enough to make sure that these people had enough time to finish everything on time, taking in account that anything and everything can (and probably will) go wrong.