Yes it is common, just not always to the scale of a Veritasium video. Usually it’s just the press office for a university putting out a press release or a summary article in Scientific American.

But in the case where the story is interesting to a larger audience, having a push behind a story across non-academic media is not unheard of. If you can get some media coverage of an academic topic, it can be very beneficial to the researchers’ careers. One goal for a researcher is to bring notoriety to their research, to their institution, and to the field in general. This is the main motivation I see.

The authors may have pushed the arxiv paper out earlier due to the timing of the release of the video.

It doesn't sound that conspiratory: someone of the millions of subscribers of Veritasium watches the video and thinks that is good content suited for HN, but instead of linking the video itself (rarely video links raise upvotes here) it links the primary source.

But even if the paper was "promoted" (i.e. a link submitted), what is bad about it? People seem to find it interesting, and unless there are upvoting bots involved, posting links is the raison d'être of this site.

I doubt it based solely on that there are multiple interviews including from one of the paper’s authors. Given that Veritasium is a very well known channel at this point I wouldn’t be surprised if they were contacted instead and then roughly coordinated the timing of the paper and video release together.