I watched the videos again several times and there are different things going on. The initial bog is clutch/rev control, but as you pointed out, there's something else going on. It absolutely dies and it isn't subtle. It's like you lost sparks for a moment or two. Sounds like my last runs where the kill switch was having problems.
I would start by testing the air over electric switch with a meter and blank off the tube to the shifter to save the forks. It may be sticky or it may be the push button that's sticking and not releasing fast enough.
You suspected fuel supply. I'm not convinced that's the issue, but check it. Hang on a minute. Where is the fuel tank mounted now? Is it possible that fuel is sloshing rearwards under acceleration and starving the carb? On mine, I extended the (aluminum tank) at the back so that fuel is pushed back into that and it can't starve the carbs. Same idea as a TZ tank but the sump is tiny. You thought about moving the tank forward at one point. That sounds like a good idea. As long as it's ahead of the carb, it should be OK.
You can also test the fuels supply to make sure it's adequate.
At this stage I wouldn't worry about where you shift into top gear because when you get a clean run, that will all change and with a relatively wide ratio transmission you could think about leaving in second, but it will put more strain on the clutch. Keep the faith. The solution is close.