^I think the mods essentially do have a part-time, unpaid, voluntary job to keep this site running (see GJ's item 4 in the email above). And it's appreciated.
There's another angle to why mods writing up birthday greetings wouldn't be the best approach after looking at the latest emails - it could present a conflict. Imagine a scenario where a member's been placed on warning by the mods and then their birthday pops up and the same mod writes up a happy birthday thread. Seems a bit akward.
Concept withdrawn from the table.