If I were to tell you that I have made two very similar videos and backlinked one on purpose I would be lying. I have done some mini case studies on my own channels though. For example, I had a video ranking on the bottom of page two for a very popular search. (It got 20+ views a day from searches near the bottom of page two without a custom thumbnail) I then posted that video to a couple of places creating back links, and it was ranked near bottom of page 1 a few days later for the same search. This video had been public for over 2 months. I would consider that a successful case study as the video now gets more than double the daily views, and it is slowly ranking higher. I actually just changed the description of it and added it to a new playlist, so I will have to see how that effects it soon. I have had numerous small "studies" like this one on my channel that have shown a positive correlation between search engine views and backlinks.
Well, the first problem here is, that it's unlikely that the backlink was even indexed that fast. When doing backlinks for Google, people expect 2-4 weeks of indexing time. Unless your video received a lot of traffic from those links (which you could just check in your analytics), it's extremely unlikely that backlinks would affect the ranking of the video "a few days later". You could also throw the video in Ahrefs, and see if it picks up those backlinks at all.
Next, lets assume they were good quality links, and people were coming through them: did those people comment on the video? Rate it? View it for a long time? If so, any of those would be a much better indicator of why your video ranked higher, than the backlink itself.
It's like testing a new drug, but not having a placebo group.
Next of course is the issue of the ranking itself. The ranking you see, is most likely not the ranking I see. The ranking I see, is most likely not the ranking a random user from here would see. YouTube personalizes results much more than the Google results (or, at least, does it in a more hidden way), so it's hard to say if the ranking increase you saw, was a general rank increase at all. Not saying it's not, but I'm saying that checking from your own computer, isn't a viable way to tell if the rank changed or not.
Last, of course, there's the Google Dance, which exists even more so on YouTube. Rankings change all the frickin' time, for no reason whatsoever (observable to us). New videos come out, old videos gets taken down, things trending, and whathaveyou. If this was done on a website, saying "I made this backlink, and a few days later, my site had moved up" that would count for nothing. Small changes happen all the time.
Now, I do want to make this clear:
I'm NOT saying that genuine, great, traffic providing backlinks mean nothing. I do personally believe they do. But I have yet to see a single shred of half-decent evidence to actually support that, let alone support that they have a big impact on rankings. In fact, everything I see point in the direct opposite direction. Which makes sense, YouTube is a social site, it puts weight on very different aspects.
So yes, backlinks are certainly relevant to your video. Or, at least, they can be, if done right. At least I think so - I mean, after all, I have nothing to back that up. I can back up the opposite though. But it is my clear belief that for a backlink to have any significant, measurable, impact on your videos, it needs to either provide you with decent traffic, or come from a very high authority site (which would probably provide you with traffic anyway).
I've recently redone the way I use my tags and I'm still tweaking them.