Not every game has to be better. In fact, games from that era looked better on the PlayStation 2, probably because the console's API was more advanced than the old DirectX that was pretty simple and couldn't replicate the cool postprocessing effects that the PS2 had. Good examples of that are Silent Hill 2 soft shadows and Silent Hill 3 advanced texture effects (search for a comparison between the PC and PS2 of the car scene and look at the windows). And in some game ports the devs might have screwed more than just textures or they couldn't correctly port them for some reason or another. But I've explained the process before, so it's pretty clear why some versions have better texture quality than others.
And for upscales, just look at both Legacy of Kain: Soul Reaver and Resident Evil 4.
- For the Legacy of Kain remaster Raina had to use the Dreamcast version which is superior in graphics and has more framerate than the PSX version.
- For the RE4 remaster the team had to use the Gamecube textures for Leon's hair since it has more resolution and quality than the PS2 one.