Why couldn't it be just a weirdly energetic neutrino originating from the neighborhood of some black hole?

At some level, it could be (and that would be an great discovery as well!). It's a question of probabilities: it's unlikely to be any of the things that we already know about, but that doesn't mean that it's something new. Unlikely things happen --- infrequently. As stressed in the article and elsewhere in comments, more data should elucidate what is going on. That's the difficulty of these kinds of searches: there is one event, and we can't make clear, confident statements about one event.

Based on my very amateurish skim of this and a related paper, maybe so? They simulated the neutrino background as solar and cosmic ray atmospheric sources, so any source with a different energy distribution is perhaps a possible explanation for the event, I think.

But probably more likely is for this to have been a particularly energetic event in the tail of one of the known sources of neutron recoil detection they did model. More events needed!