About the video itself I found some threads on r/askhistorians:
What I've gleaned from these posts is that the archery depicted in the video is only a separate style, but not the one true style archers used in the past. His style of archery isn't a "forgotten art" so much as it wasn't even an art many European medieval archers used to begin with. That is, the facets of archery that the video claims are "myths" aren't actually wrong; those myths were true for some types of archery but not others. So the modern perception of archery (what the video tries to debunk) isn't necessairly inaccurate as a whole. Remember that technologies and techniques tend to develop differently in different areas of the world; the archery in the video seems to have been mostly used by non-European (horseback?) archers, and that European archers worked the way you would imagine. I don't know. I know very little about history, having never taken a proper history class. It just seems from the Reddit links that the claims the video makes are exaggerated, but not false (originally I was going to write that the posts said the video was all inaccurate, but then I realized that the posts that said so were mostly written by one person and they stuck out a lot in my head.)
Still an interesting video, in my opinion, but maybe the tone is too... click-baity? Promotional? "Cool?" It's hard for me to describe this sort of thing.
What's the takeaway for Fire Emblem, then? Maybe there could be more different archer classes, some of which use a type of archery like that depicted in the video, and some who act more like stereotypical Fire Emblem archers (i.e "modern" archery.) The former group could get a skill that allows them to attack at 1-range, as other posters here have said. The other could... hm... what are the advantages of stationary archery... have increased accuracy and critical rate? More sheer power? A fragile speedster vs mighty glacier divide seems logical, I guess, except the stationary archers might have increased range instead of having good defences... They would have three-range combat (or more?) in exchange for not having one-range combat. There's obviously a lot of misconceptions about medieval combat and Fire Emblem probably uses plenty of them (weapons were almost never heavy enough to weigh the user down in real life, apparently), I suppose.
Also... what if the "modern" archer class could attack at ranges greater than three, but couldn't double and/or had reduced damage? That might be interesting... and I think that might simulate uses of archers as artillery units better.
Don't know what to say about crossbows. The video doesn't actually say anything about them and I have little knowledge of them.
... Incorporating ideas like this into Fire Emblem is fun, I swear, at least as a theoretical exercise.