Otherwise, the PS1 is known to have pretty bad dithering in some situations, so it can look pretty bad when scaled up.
Yep, this. Some titles – such as Gran Turismo and Silent Hill – exhibit that quite glaringly, especially when you switch from CRT screens to current displays with scalers. It comes out extremely obvious.
Also. Are you using an actual PlayStation to run PS1 games, or your PS2?
I’m asking this because I noticed – not investigated it yet, due to the lack of the necessary hardware – that my Japanese (fat) PS2, even with all the back-compat. enhancements turned off, seems to introduce some very odd artefacts to the PQ of some PS1 games – Tekken 3 and Tobal 2, to mention the first ones popping out of my head – similar in many ways to a ‘grid’ pattern.