That aside, I'm also an amateur, so I know what it's like trying to chase down these things!
It might be worth having a close look at the schematics, and working back from the socket to the shifter and beyond to see what points there are where interference can be picked up. I'm surprised that the caps suggested by exxos had no effect, but what I would say is that it does look like the banding on the old Spectrums et al that was caused by poor filtering, I think on the power to the RAM.
I know it's not quite the same, but it may well be worth testing by adding additional filtering to the RAM power lines, and considering a more general re-cap of the board. Generally the caps do seem to be ok on Atari's though. Are you in a position to try with a different PSU? Or a bench PSU to see if that eliminates the problem?
A couple of other things. Have you checked the video socket and reflowed? Have you tried another video cable?
Finally, I'm not sure what other video options the MSTE has (I only ever use mine on VGA), but it may be worth checking what your options are, such as SCART and composite to see what the results are like there. I think you also mentioned you use an OSSC, and I have found that some things pick up weird interference problems when I use mine, so I have a select group of items that I pump through it. It would be worth trying to use it without the OSSC to see if that makes a difference.
