This isn't a particularly controversial claim; if you've studied archery, it is unexceptional. I've seen the same argument in multiple places, generally with evidence very similar to what Shad presents as his own research. I believe the same claim, with much of the same evidence has been presented on Modern History TV and on Malta Archery. I think Nusensei may have also addressed the topic.
If I get time, I'll try to find the individual episodes; the point being that there are other sources to consult. Aside: An acquaintance of mine did some research on Shad; suffice to say that Shad's enthusiasm is more compelling than Shad's research/qualifications. I think that Shad's video generates the impression that Shad's research is novel/innovative; I don't think it is. I think Shad is attacking a straw myth - something that is not believed among scholars of the field.
As a horse archer, if I draw Mediterranean, my arrow rests on the left side of the bow. (speed load - slap the arrow on the string, thread it through to the left. Arrow rests on the left side of the bow and the right side of the string as you draw your hand back, allowing the string to slip by your fingers and guiding the string into the nock. All of which must be done without taking your eyes off the target and while in a half seat on a moving horse. At all points I have complete control of the bow and the arrow. Because the horse is moving forward, wind pressure is trying to push the arrow off the left side of the bow, so I have to maintain control. I can get off an arrow about every six seconds; I think my instructor can do every 5 seconds or less at a gallop.) To address the potential geographical question, horse archery is extremely popular in Turkey, which is arguably part of European history during the period.
If I draw with a thumb ring, my arrow rests on the right side of the bow - different style, but the same outcome. In this case the wind pressure holds the arrow against the bow - but thumb ring AFAIK, less common in historical Western Archery. (ASIDE 1: I haven't trained myself to do Persian, but I believe it is also right of the bow, as is, I believe Slavic and Apache).
The point is that different styles do different things. There is diversity among historical archery. (Malta Archery is a very good source on this, exploring Mediterranean, Oriental, Persian and Slavic draws among other things). Mr. Hirmer makes the point that our ancestors weren't idiots - they experimented with a variety of techniques and preserved and used what worked.
OP asks why the right of the bow became "normal". I don't have an answer to that. I have several hypothesis, but I don't have enough sources to validate any of them (could be an accident of the traditions that got preserved vs those that are lost - common in any martial art. Could be that right side of the bow is simpler to learn. Could be a combination - that left side of the bow is more practical for stalking, but right side for target shooting, and history privileged the target shooter. ) Testing any of these would require more sources than I personally have. I'm rather skeptical that we have enough primary sources to reach a reliable conclusion. If someone were interested, I'd reach out to Joe Gibbs, or possibly to Tod of Tod's workshop. Both of those individuals have demonstrated a greater respect for historiography than is present in the referenced Shad video.
I'm not expert enough to make the argument here, but I would love for someone more skilled in "living history" as historiography to comment. There are some challenges in studying the history of behaviors, practices, crafts and traditions that weren't associated with textual artifacts. I won't pretend to have enough of a background, but I think every historical research into archery I've seen consults the same handful of source materials.
Actually, this may be easier than I thought to answer; when the fletching passes the bow, it leaves a mark on the bow. On my primary bow the leather wrapping on the left side of the bow is completely destroyed and there is significant damage to the fiberglass. (I shot Mediterranean for a year before I shifted to thumb ring; the wear is much heavier on the left). I don't think we have a huge inventory of historic bows, but I know there are some recovered from the Mary Rose (with draw weights that are incomprehensible to the modern archer)- reviewing those should indicate which side of the bow the arrow passed.
(I'm making a heroic effort to suppress my desire to include my "History without sources" rant here.)