The degree to which this technique works definitely does depend on the rim/tire combination. On my Friday I have an aero Sun rim on the front with an IRC 1 1/8" tire and on the back I have a non-aero Sun rim with a Primo Comet 1 3/8" tire.
The rear tire is ridiculously easy to get off the rim - just deflate, grab a section of tire and pull and it comes right off - no tools or special technique needed. OTOH, the front combination is much harder. Even though I've already gone to thin tape replacing the rim strip and use the technique shown to get the bead into the center of the rim I still have to struggle a bit with tire levers to get the tire bead over the edge of the rim. My other rim/tire combinations all fall somewhere in between these extremes. I have always used the technique shown to make it easier and am usually successful to getting the tire off without tools but there are exceptions.