I take issue with your translation at only a single point:

I'm making progress then. :)

When I follow the same procedure—gather all the quarks—I will be equally justified in gathering a set and in gathering a superset consisting of one other quark.

No. If what you gathered is a proper subset of what you could have gathered, then you didn't gather all the quarks, and you're not justified in claiming that you did. How did you decide to leave out that one other quark? You must have made a distinction between it and the others that you did gather.

There's no way for me to distinguish the two sets.

Of course there is. The superset contains a quark that the subset doesn't. If you refuse to notice the differences that single that quark out from the others, that's your loss.

I think that maybe you're trying not to distinguish between quarks, but are implicitly distinguishing between "quarks that you know about" and "quarks that you don't know about." So you might assemble all the quarks you know about--an infinite number--and not have any evidence that this isn't all the quarks there are. But later, you worry, you might find some other quarks that you didn't know about before, so that your original set didn't actually contain all quarks. This is not contradictory. If there was a chance that there existed quarks you didn't know about, then you weren't justified in saying that you had gathered all the quarks.

following the procedure "gather all the quarks" should constrain me to a single set, "all the quarks," rather than allowing a hierarchy of options consisting of supersets.

It does. If you're not at the top of the hierarchy, you haven't gathered all the quarks. And you can't justify claiming that you're at the top of the hierarchy by blinding yourself to evidence that would prove otherwise.