I think you're right—and, unless you wash the bottle by hand before sending it to the recycling center, that's not a "tiny nitpick," it demolishes my whole thesis. But maybe the "thesis" was already so full of caveats as to be fairly demolished in the first place, unless you reduce it to "it depends".

Recycling waste water is a whole science, this goes for car washes, dishwashers (which now use less water than doing the same thing by hand) and industrial washing of all kinds of materials. The times when water was used just once are long gone, and that's before we get into thermal recycling to ensure that the maximum amount of energy is extracted from whatever waste remains.

Do you have a handle on how efficient these systems are and how much energy it really costs to reuse a glass bottle? Or how to find out?