I would not specifically say that, logic is formed quite specifically you can tell from weak signals to dirty signals all with a scope I would easily use it over a probe every time...
Since its background scrolling I would maybe (without looking at any schematics at all) find the ROMS that hold the data and work from there... Check the ROMS are all good and the sockets are 100% (press on them or pry them a bit while the game is running and look at the screen)
Also you may want to look at the shift registers (usually 74LS164, 175, 166 but may be others) https://en.wikipedia.org/wiki/List_of_7400_series_integrated_circuits
And the last thing to check in this area is any ram dedicated to the background, of all things ram is often the only component complex and dynamic enough to cause randomness in errors where as specific logic always faults the same way.
Have a look at mame and see where the ram regions are or the locations of roms etc, check for damage or corroded tracks or components bent over etc...
There are many other tricks to try, you can piggy back spare ICs or use a pull up or pull down probe to force specific inputs or outputs and see the reaction on the screen...
Anyway there are heaps of possible things, the chances of fixing are in the details of the fault, your description is just not specific enough for anyone to really say much about it, how a fault behaves onscreen as well as the shpe, and changes that happen when cold goes to hot etc all play a part in diagnostics....
Video is the best way for others to at the very least set you on a closer path to glory ;)
Good luck....