I've been heavily in the decision process of rejecting interview candidates at my firm. We have certainly come across PhD candidates who failed sheerly based on the poor quality of their presentation.
The application process is an art. It's a crapshoot. Even to your recruiters. Sometimes we already have a hypothesis against the candidate's favor by the time we offer him/her a chance to work on a 1 week take-home project. For example, a candidate sounds wonderful on paper (statistics PhD, list of programming proficiencies), but spams us with a list of 30 'papers' on his resume which are really just a mix of reports, undergraduate theses, technical white papers, meta-surveys of the field and conference summaries.
At this stage we already suspect he/she is the type to take credit for anything and don't really want to work with such a person, but nevertheless choose to administer a take-home test. Why? Because the marginal cost of administering a take-home test is nearly zero, but it gives us a chance to confirm the flaw we're rejecting him/her for. Perhaps his/her paper ends up with a working model for the problem, but say the tone of the conclusion suggests a much more important result than he/she has proven, it's already enough reason to reject the candidate because we have a vague confirmation of our earlier hypothesis.
(Don't hold it against your recruiters. False positives are much more costly to the firm than false negatives and most firms that have spent >2 years' time at this would have come to the same conclusion that they would much rather hire a less skilled candidate who is less likely to give them interpersonal drama in the office.)
It's the classic noise and signal problem. Basically, so much can happen behind the scenes (noise) that it's not possible for us to speculate based on your post. Your recruiters don't want to give any reasons for rejection that may be misconstrued or misrepresented as an unlawful, discriminatory action in court, so there's plenty of ambiguity (weak signal). The best you can do is to ask their HR rep nicely for some feedback.
As for examples of acceptable presentation skill, I imagine a MCM quality solution would have been sufficient for a 1 week take-home. (This is also why firms such as Jane Street and Two Sigma particularly respect candidates who have performed well at the MCM.)