I’m not sure I agree. I assume there’s some text missing in the email thread, but by itself the email doesn’t mention that the deadline is supposed to be a one week extension. It only states 15th August. I don’t think it’s reasonable to argue that the applicant should have realised that the date was wrong when the university should have sent the correct date to begin with and avoided confusion. Surely the burden is on the university to be clear about their requirements. Also, I note the reason why the applicant needed an extension and therefore their priority might not be spotting errors in correspondence...
However, I think realistically it would not be worth pursuing the matter - your energy would be better spent elsewhere. I’m sorry that such an unfortunate thing has happened to you though.