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 is giving very physical and "natural language" arguments. You might find a different book (Landau and Lifschitz?) that is at a higher level of mathematical rigor less frustrating.

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.