Usually you like to start with the easiest solutions. That being said, are you sure the cart slot is clean? Some times with these older consoles, depending on where and how they were stored, the cart slots will be VERY dirty and possibly even have corrosion.
Here's what I like to do: Use a piece of cardboard or a few thick magazine pages folded together or a cereal box folded together to get the desired thickness (perhaps a little thicker than a SNES PCB). To thick and you can damage the pins. To thin and you wont be able to scrub them enough. Wrap a clean white piece of fabric (like an old white T-shit - something that wont break apart and leave debris in the cart slot) around it, dampen it with some alcohol or electrical contact cleaner, and insert it into the cart slot. Move it up and down several times. Remove it then look at it. If the white fabric is dirty, then you know you need to clean it more. You may have to repeat this process (moving to a clean piece of fabric) several times before it's completely clean.
Put a flash light on the cart slot and see if there's any corrosion in there.
Also, don't forget to clean the smaller outer pins. Star Fox and Vortex both have the Super FX chip which uses those outer pins. You wanna be sure they're clean as well.
And it might go without saying, but make sure the carts too are as clean as possible.
Assuming everything is as clean as could be, there might be a problem with components in the console. I've seen similar problems before, with speculation of the PPU or video RAM going bad. My first SNES (from 92 or 93) has a similar problem, although not nearly as bad. I figure the PPU or RAM is going bad, after reading of similar problems online. If this is the case, there's not much you can do.
Star Fox and Vortex both use the Super FX chip, and F-Zero uses Mode 7 heavily, so these are three games that I guess you could say are graphically taxing for the console. Have you tried any other games, more "normal" games? Something basic like Super Mario World.
I'm not sure if bad capacitors could cause such problems, but I suppose that's another possibility. When you cleaned the inside of the console, overall, how did it look? Maybe take another look inside and see if anything stands out as wrong (bad caps, corrosion, anything obviously damaged).
If you don't already know, you can spot bad caps when the tops of them bulge or even leak/explode.
When they start to barf (as Hobie likes to put it
) that's easily noticeable, but also look for any tops that are swelled. And also, it kinda sucks, but you can have bad caps but not be able to visually see that they're bad. People often "recap" old electronics, by that I mean replace every capacitor, but that's kinda a pain in the ass to do if you don't know if that is indeed your problem. Unless you see visually bad caps, I wouldn't know how to tell if that's a cap problem or something else.
I'm sure there's other things to suspect, but if it came down to bad RAM or PPU, those aren't parts that you can just go out and buy. Basically, the only thing you could do is salvage those parts from another broken SNES (broken in another way, of course). And even then, replacing them requires advanced soldering skills.
All that being said, the first thing I would do is be sure the cart slot and carts are 100% clean.
