For those well-versed in physics, it seems mind-numbingly bizarre to hear someone claim that the Moon's gravity isn't enough to affect a pen, but is enough to affect people wearing heavy boots. But as for some hypothetical person who hasn't studied much physics... or screw the hypotheticals - for me, this sounds wrong but not obviously and completely wrong. I mean, "the pen has less mass, so there's less stuff for gravity to affect" sounds intuitively sorta-plausible for me, because I haven't had enough exposure to formal physics to hammer in the right intuition.

Absolutely. Another piece of the puzzle required to understand whether the pen 'obviously' falls or not is, 'what kind of atmosphere does the moon have'? What fraction of people know that there is no atmosphere on the surface of the moon? (Do I really know this?? I think I just remember being told this, and despite being told, I'm not certain there's absolutely no atmosphere on the moon.)

Without detailed information about the atmosphere, you really don't know. On Earth, the pen floats in water, but doesn't float in air.

(And then you have the added problem that there's a high chance people will first recall the image of the flag blowing on the moon, which is unfortunate for physics.)