These emails are interesting for consultants going through the interview process. It’s quite wrong to treat people the way that you have been treated and it is rude. It happens in the US, it happens in the UK, in fact it is probably a global problem. It got worse in the early nineties recession when photocopiers hit their peak as the preferred tool of mass rejection, because the generic wording was applied to all applicants whether they narrowly missed getting the job, or were never really under consideration. Again in the downturn, a few years ago, people often used emails to communicate their pleasantries with negative messages, because it was more efficient(and less embarrassing), than using the phone. Sometimes people can't handle doing it - they are after all human beings - so they freeze unable to deliver a simple 'no thanks, good luck'.
It happens. If necessary have another career think, and start applying again. If you cannot answer the question "Do I really, really want to work in this role?" with a positive response, then it is a good indication that you should keep searching. It's not you with the problem, it's them. You progressed through their interview process for a while, and despite all their people and resources, you won and they lost. They might have better days, and it might have been a genuine mistake, or they were embarrassed, or whatever, but they played the role of the loser with you because they messed up at the finishing line.
However in your emails you told us how many applications you made and that you now felt that this is your only chance. It might seem that way, but it is not. Unfortunately I have encountered worse post-interview behaviour than this, and encountered less favourable application:interview ratios, but still won spectacular consultancy work elsewhere. Every consultant encounters it to some extent. Some talk about it, others don't. Even McKinsey experienced rejection when he tried to apply his business skills to running a business outside of consultancy, later in life. To say that he had a top consultancy career might be considered by some people to be an understatement.
Post-interview procedure obviously varies according to the people involved. If you go through a head hunter/recruitment agent you would do well to agree it in advance and they can help you with this. If you apply on your own, you should thank them for their time and try to obtain a response, in much the same way that you did. A carefully worded email or phone call is often standard behaviour now for this.
As you probably know if you are positive before, during and after the interview, you tend to leave a better impression, so that you can then turn them down if you get several offers. The global economy is looking more positive which obviously creates more consultancy work. You will get other offers, and who knows perhaps they will still be thinking about what to say to you! Oh, and if you don't get the job and they can't say it, then take it from a sympathetic stranger - Good luck with your future career.