Yes you were totally correct in the beginning. Orgopete confirmed you were correct. I think somebody else was confused when they proposed a birch, and/or radical type mechanism.
The fact that a molecule is a product of some reaction does not tell you how it will act as a substrate for a different set of reaction conditions.
Calling an acid base reaction an "H elimination" in your original post is an imprecise use of language, though. That could mean H+, H minus or H radical. Usually simply saying "H" is used to mean hydrogen radical. It's better to say "proton removal" or "H+ abstraction or something like that.