I’d say that defining the third stage as a payload is a bit of a stretch, but I agree with the point that defining “payload” can be tricky at times.
The Falcon numbers don’t include the fairing, which doesn’t remotely make it to orbit (it’s jettisoned not long after MECO). I don’t know about the payload adapter.
In any case, the Saturn V clearly outclassed the FH by any metric, even if we might quibble about exactly by how much. That said, the FH could easily support an Apollo-style mission if desired via multiple launches. It would hardly increase the mission complexity at all given that Apollo already required a kind of self-rendezvous during its “transposition, docking, and extraction” maneuver.
In any case, the important part of the FH isn’t so much the “heavy”, but rather the $800/lb launch price ($95M for partial-reusable mission, 56 mt to LEO). I really hope that people are thinking of new, interesting things they can do at that price.