And, of course, half an hour after giving up and asking for help, I discovered
what I needed to change.
I did a "journalctl | grep smtp" and noticed that, when my machine was
connecting to gmail, it seemed to be doing so on port 25. Aha!
So I changed my transport file explicitly to use port 587 when connecting to
smtp.googlemail.com, reloaded everything and now it works.
(Slightly in my defence, I had briefly pondered the question of port number
earlier this morning, but, since I hadn't seem any mention of it in my reading
of solutions to this problem, I figured that the fact that I had enabled auth
in the main.cf file must mean that postfix was automagically going to use port
587 instead of port 25. Now I know better.)
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