Meanwhile, so called overrated PS2 has a handful of great games, and a ton of crap that nobody remembers. Most games weren't fun on that console.
I blame the success of the PS2, it was decline. Quality over quantity, and the Gamecube was BY FAR the leader in this area, just like the Megadrive/SNES were at the time.
Sony's strategy of going for quantity dates back to the PS1 where they really did as much as possible to get third party developers on board with extremely good deals regarding royalties, disc pressing price, dev kit price, ease of development etc. PS1 has a shitoad of shovelware as well. PS2 largely continued that apart from ease of development, I think it was way harder and it it wasn't coming after the PS1's success devs would have rather avoided it.
I had (and still have) both back in the later half of their lifespans, and this feeling that the PS2 didn't have that much good stuff on it is something I can relate to (also the huge wow factor of GTA3 that sold a shitload of PS2 was not a thing in my case, as I played it on my PC first). Still I think there's a fair bit of interesting obscure stuff to find in the ocean of crap shovelware the PS2 has, more than what the cube offered in total.