Given most of your games seem to fail to fire up, does sound like possibly the slot just needs a bit more
cleaning - I used a piece of stiff cardboard and some methylated spirits on the edge a few times until the card didn't come out dirty. Took a couple of goes to get it all, but it did help bring it back.
I'd also clean the edge connectors on the cartridges as well with a bit of meths and a cotton bud.
Another thing I heard mentioned was pushing cartridges in too far. No idea if that's true, but you could try putting one in, then gently pulling it out just a little...
Good luck. I can't really suggest more then that - though I'm sure some of the more experienced SNES'ers on the forum here may have other suggestions.