Old PS1s can be pretty picky when it comes to which CD-Rs it wants to read. CD-Rs are all a bit different physically than real PS1 game discs, and the PS1 was not designed for maximum compatibility with CD-Rs. Furthermore, the age of the laser pickup diode could be a contributing cause. Over time, lasers in CD and DVD drives get weaker and weaker, making it harder for it to read stuff. This would likely start to be a problem on CD-Rs (which it is not even designed for) before it starts being a problem for legit PS1 games.

As a solution to your problem, I would suggest trying a different brand of CD-Rs, and clean your PS1's pickup laser. You could either use a cleaning disc, or do it manually with a q-tip and some sort of alcohol/white spirit/isopropanol etc. Just wet the q-tip with it and try to wipe the lens.