Thanks for the reply. I found the problem, strictly by luck.
The mozillaZine Knowledge Base has a goof file about transferring files
and folders from an existing Firefox profile to a new one.
I had been copying the files, and then the extensions folder. When I did
that, as soon as I restarted after copying the folder, I was presented
with a lengthy set of tabs, at least one per extension, asking
permission to install it, a tedious procedure at best.
Yesterday, in trying to find another missing extension-related icon, as
the extension’s author happened to reply first, I created a new profile,
and before doing anything else, I installed the extension. To my
surprise, the previously-missing icon was present. I then copied the
extension folder into the new profile. When I opened Fx, instead of that
set of tabs, what I had was a complete set of disabled extensions, so I
started enabling them in sets. Among others, the icon for Flash Control
was then present, and as I progressed in setting up the profile, I came
to realize that several icons were missing in the profile from which I
copied the folder, a profile that I had waded through the tabs when
setting it up.
Case closed, dismissed with prejudice, the site (your extension)
returned to the owner (something I did more than once as both a
volunteer and career fire chief.)