A friend of mine has a bike with a certain rim/ tire combo that is a real bugger to get the tires on the rim. She has Pedros tire levers, super super strong. I bought myself a pair just to have around for tough cases, and they won't break like thinner ones.
One trick I always recommend is the old pedal strap technique.
Look up on YouTube, how to get marathons on a wheel, something like that, and a British guy shows it. Really makes the difference with hard to get on tires.
Here's the link, will help immensely for hard to get on tires, just remember the concept, it works.
Tire bead pushed down into well of the rim, gains you that little bit needed....
https://youtu.be/-XUFVrl0UT4
The method you recommended for mounting tires is excellent. My wife and I taught a bike maintenance course and she would demonstrate fixing a flat tire. She could mount most tires onto the rim without the use of any tools using the technique described in the video. For most tires the pedal strap is not needed, as long as the tire is pinched toward the center of the rim , and pulled forward as you work around the rim from both sides.
We did have some Continental tires that were really difficult to mount. I was not sure whether it was the tires or the rims we were using at the time. We also did not know about the method shown in the video at the time.