Look at the link I posted above. It's in there. This topic always gets people in a foam and it always has. I get it. Many people will be skeptical and many people would rather not reexamine what they've been doing in the past. I'm not here to convince anybody of anything. Try it or don't try it... it's that simple. There are hundreds of brewers on that low-O2 forum doing this in one form or another. For it to be confirmation bias it would have to be every one of them. Some of the people over there are just humble beermakers like me but others are engineers who have more scientific credibility than I do.
That's pretty much how confirmation bias works in an echo chamber.
I couldn't see the pictures of the transfer on my phone or I didn't scroll down enough. On the computer now; I see it the transfer makes sense. Just have to have enough pressure in the keg to get all the beer moved over prior to equalizing the pressure. That's kind of neat. Again, there's no reason why you couldn't use a siphon though, just put a valve in the line on the hose to prevent spillage during fermentation and do your process as normal. I'm just trying to help with your issue of ingress to the fermenter in a swamp cooler.
The real solution is obviously get a bigger fermentation fridge with multiple temperature control chambers so you don't run into this problem again.
O2 ingress aside, I've always thought through my own anecdotal tastings that beer that I've brewed that naturally carbonated tasted different, sometimes better, than it's force carbed counterpart. The biggest problem I have is that there are hundreds of variables to control to really give a good comparison and outside of a lab it's so hard to make a definitive statement. Do what your taste buds tell you to do.
You're mash cap needs better materials. Aluminum oxide on the foil will leach into the beer at mash PH.
Also I'm really surprised you haven't switched to BIAB yet to avoid sparging.
