Do you think it's more likely my PS2, then?
I would probably work on a ps2 with a good laser yes.
However in said state a clean copy of X would probably be read withouth issues.
In your state I wouldn't go for drastic measures if all other games can be enjoyed however. Take it slow no rush.
FF X is a Very cheap and common game, it's not something that even resellers can get rid of easily. you'll find a cleaner copy easily at some point on the cheap.
Get a near mint to mint disc in the future if you decide to try it out. aka only very few very faint scratches if not better.