For example: let's say that you're speedrunning ancient cheese and the high score is around 70,000 because of a mod before, but you're playing on an unmodified version at that time. Would the run still be allowed?

Past high scores shouldn't invalidate a new run, as long as the run itself has no modifications, if I understand correctly.

Yeah, as long as the current build you are using to run is not modified in anyway, that's ok. Save data is not part of this rule, as it doesn't change fundamental assets, like music, sprites, or movement.