Usage of "though" (was "want")

"I went with him to the fountain, though not thirsty."

I'm surprised that four people find this not quite correct, since it
works for me. Could it be an American thing?

I've just checked the OED and they list the "parenthetical subordinate
clause with ellipsis" as going back to LME, with a 20th century example
from Barbara Pym.