I looked through the PS1 store online, and didn't find a lot I was interested in, except for some racing games. The graphics are dated, but I'm sure the gameplay could be good, especially if a game was the 'best' for its time.
I am chuckling at your non-violence question, as I didn't really want to explain it but that's ok lol.
I've just thought a lot about morality and the idea that doing something simulated sort of trains the brain to do it. For instance, after playing a lot of Grand Theft Auto: Vice City as a teen, I felt that in public I had a more hostile reaction to people than I did before the game. I've also read that a small part of the mind believes what it is seeing. Though there is no way to prove this, it still is slightly scary to me to think of doing simulated immoral acts when one is trying to be moral in real life, seems odd.
So now I'm taking it to sort of an extreme and not playing any games that do violence to a human or animal, with the exception of something so cartooney or with graphics so simple that it isn't realistic at all. For instance HOT PXL has a mini game where you use a godzilla foot to stomp on a chick, but it is done in a very funny way and not the same as say, shooting a few hundred people in a realistic first person shooter game. I'm not preachy about it and don't expect others to avoid violent games, It's just something I'm personally doing. I love retro games, puzzle games, really I've always preferred the other types anyway.
This means that I skip past all the top rated violent games, as gaming seems to have an obsession with violence (I think critics rate violent games higher, look at any metacritic scores, you may see what I mean), but there are still a lot of titles. Sports games, puzzle games, and racing games mainly.
I'm also even more picky and don't like any games that look like they're for little kids, so I tend to avoid Nintendo entirely. This leaves me with comparably few options, but there always seem to be more than enough titles to choose from.