I'm not sure how a gag strap would have helped the situation, and if anything, I could see how it could have hampered the situation.
Right. I think the claim is that a gag strap is helpful in the event the diver goes unconscious. I'm not sure if there is anyone suggesting that it's helpful when the diver is conscious and keeping the mouthpiece in their mouth without help.
In the scenario that happened to you/your buddy, if your buddy had had a gag strap and the same training I did, then (hopefully) he would have closed the DSV (as he presumably did) and pulled it out of his mouth and down onto his chin in order to then allow himself to get his BO reg in his mouth. One the DSV is pulled out and down like that, I'm not sure if it would really be a problem or in the way after that, even for people like yourself and your other buddies who were managing the hypercapnic diver. But, I don't really know - meaning, I haven't been there. I haven't seen it. I only know how I was trained and, well, what I read on the Internet...