If you want to solely devote yourself to FF7, you might be able to get through that. I had a PS1 with a broken door sensor so that I could play with the lid down, up, or the top cover completely removed. I got to disc 2 with no memory card iirc.
As for street fighter games, a lot of them have "unlockables" that would kind of suck to lose from no memory card... I think you might be better off simply emulating PS1 as I believe they spin at different speeds than PS2 games and you can mess up your system playing them.
