Now, I'm not an expert tire roller, but I have quite a bit of process flow experience. Do you have enough grip surface/angle to get your feet closer to the tire? It looks like you can get more hip & leg drive if you do. If you can gain monentum at the bottom, you can sort of "clean" the tire up, & flip your hands around. Walking the tire up is causing you to have to use your elbows & forearms when you transition your hand placement, cause the tire is nearly stopped. Easy for me to say, right? That's one big tire.
After the tire comes down, you reset your hands & chest on top of the tire. If that's not required, it could cost you 1 second per flip. (I don't know if you're making sure the tire has stopped moving). You could go straight to the base of the tire again, butt down, & blast it off the pavement. Power certainly isn't your problem. Using the video can help clean up any wasted motion. Economy of motion equates to economy of time & energy, so you should actually have more energy to flip the tire more times in less time. (or something like that).
Anytime. Look at the vids again, you'll see what I mean. I looked at them again, too. You might try putting your hands inside your feet, sumo style. Kinda like how the Highland guys throw the caper. I don't know how much grip will be affected. Again, I've never tried to flip a tire that big. It's hard enough doin the tires from my daughter's Accord.
You'll have to see what works for you.