Behold the sarcasm smile. Naturally, I did all that before coming here with my question, and was unable to find any information about an actual extension that showed actual message info.

As it turns out, the extension I was using actually does this thing I wanted it to do - what it doesn't do is in any way tell you that it will do that, and so the only way to discover as I did that this is so, is to wait until there is an actual message to notify me of, at which point the message "there is 1 message" appears out of nowhere, which is gratifying but frustrating after spending a couple of fruitless hours trying to find something which actually advertises itself as doing this. So a happy ending, peppered slightly with ennui.
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If it's brokenless, don't suffix it...