How to mount tires without tire levers?
That video is of a relatively supple tire going on easy. It's pretty blasé about the difficulties that can be encountered. He does end at the valve, which is good because advice to do the contrary is one of the most parroted falsehoods in cycling, but he doesn't talk about why.

There is a universal truth of difficult clincher tire mounting situations, and it's simple. You get the last part of the bead to go over easiest when on the whole rest of the wheel, but especially 180° away, the two beads are pinched together down into the well of the rim and kissing each other. And you must finish at the valve, because the valve is the spot on the wheel where the beads can sink into the rim well the least. 

You can use force, tools, water, soap, pastes, and thinner rim strips to help accomplish this. Sometimes even strong, skilled mechanics need or choose to employ some or all of those. It is true that oftentimes brains and dexterity are enough, but not always, and not always on an average time expenditure basis either. But the point is that if you want your maximum chance of doing it with hands alone, you have to focus your efforts on the above principle. It's how clincher tires work and there is no trick that's anything but a rephrasing or play on it.

A side note: Sometimes you hear cyclists and mechanics advising against finishing the job with tire levers or using them for installation at all. The basis for this is that doing it carelessly is a good way to ruin tubes and sometimes tires. Tubes when the tube is being reefed on directly by the lever without the user noticing or caring, and tires when excess force is applied when the beads haven't been properly sunk into the rim well, as above, and instead are stuck and won't simply be coaxed in with force. These situations are avoidable, although some tires don't make it easy. I install a lot of 25, 28, and 32mm Marathon Pluses and I usually finish with a tire lever because on balance I find things go smoother that way, but it requires being very careful that I'm not committing the above mistakes.