First just a comment on the wording: Feynman says you need to remove "a little weight," not "little weights." I picture this as using a knife to shave a small amount of mass from one of the cubes. Of course the details of how you remove the mass doesn't really matter, but I just want to make sure it's clear that Feynman is not saying to remove an entire box from either scale.
A second overall comment is that I think you are reading Feynman at a very high level of precision, whereas Feynman tends to use very physical and "natural language" arguments. Just as a suggestion, you might find a different book (Landau and Lifschitz? Kleppner?) that gives more mathematical details to be less frustrating. Of course Feynman has a lot of insight and is worth reading, but I'm just raising this since this is one common reason people find Feynman to be difficult to follow (until you already have some background in what he is talking about).
Anyway here is how I interpret Feynman's words:
- "lift off": remove some mass from the balance.
- "actual lifting machine": one whose pivot has frictional forces that act on the balance, which dissipate energy.
- "add a little extra": add some extra energy (not mass).
Putting this all together, if you "lift off" (remove) a little (infinitesimal) mass from the left plate in your figure, then the balance will start to lift the three masses. However, if there is friction in the pivot of the balance that can dissipate heat (which there will certainly be if this is an "actual lifting machine"), then the balance will slow to a stop before the masses reach their maximum possible height. Therefore one must "add a little extra" energy (beyond simply removing an infinitesimal amount of mass) to overcome friction and raise the three masses.