Feynman's layman proof of local charge conservation

I think you've confused global and local charge conservation. The quote you included actually skips parts of the video, and I think the video may actually have been edited as well, so it is not obvious from the text alone what Feynman is referring to when he says "the second form of charge conservation...". He's actually referring to the first sentence. The correct definitions are:

A quantity in a box is conserved globally if the amount of the quantity in the box never changes. This allows, in principle, for the stuff in the box to instantaneously zap from place to place in the box, so long as the total amount never changes.

A quantity is conserved locally if the only way it can leave a region is by moving with some velocity through the walls (no teleporting). We say there is a "flux" of material through the boundary, and the amount by which the stuff inside changes is exactly accounted for by the flux entering or leaving the boundary.

So your statement "if charge pops up somewhere it must have disappeared somewhere else" is global or non-local conservation, not local conservation.

I think this should clarify the argument. If a quantity is globally conserved but not locally conserved, then some observer with a relative velocity will see it violate global conservation as well. The lesson is that given relativity, local conservation is necessary to have global conservation in all frames.

I'm not sure whether everything's clear from your description and from Geoff's Answer.

If you haven't already worked it out, the essential problem that Feynman and Einstein are getting at is the relativity of simultaneity.

In the second situation, which Geoff's answer explains well, a relatively moving observer will not see the disappearance of charge and its reappearance elsewhere at the same time. There will be some observers for whom there is a nonzero time interval when there is neither the disappeared nor the reappeared charge present, and other relatively moving observers for whom both are present at the same time over a nonzero time interval. For all relatively moving observers, the charge becomes time varying.