PS2 playing PSX games was an incredibly awesome new feature. The fact that no other console had ever done that before was one of the selling points
Almost. The Atari 7800 could play 2600 games. The Genesis could play most Master System games with an adapter, as could the Game Gear. But the Genesis and 7800 got their back-compat the same way the PS2 got its: by using most of the previous console's hardware as an I/O coprocessor for (say) running audio. The Nintendo DS would end up doing the same thing.
And of course a confounding factor in all of this is the resurgence in PC games, mainly due to availability via Steam
There are still several genres that lean heavily toward consoles, such as sport games, fighting games, party games, or anything else with shared-screen multiplayer. It's common to see games in these genres released on Xbox 360 and PS3 with no PC port or a 2-year-delayed PC port because of publishers' fears of widespread copyright infringement. I guess I'll have to wait and see if the Steam Machine fizzles the way OUYA did.