BTW, Tolivac, I was looking at the "roof" and I don't think it's much weaker than a standard flat roof of a few decades ago. For example, the stringers on this pergola are 30" apart. On the roof of the main house, the rafters are 24" apart, and same size, 2x4's. Then on top of that there are spaced 1x4's with sizeable gaps, and on top of that, heavy cedar shake shingles. I figure the 2x2's have about the same effective strength as the 1x4's, and 30" is not that much more of a stretch than 24". Granted, the pitched roof thrusts a good part of the load downward to the walls, but it's a minimal pitch (4 in 12). Tying all the 2x2's together to the 2x4's below them with decking screws probably serves to distribute the load across multiple beams and helps to resist at least one person traverse.
One concern I have is the additional weight in the winter when the 2x2's get soaked by the rains. But then the cedar shake roof also absorbs a lot of water and it doesn't seem any worse for the wear (other than that it's about 35 years old now).