@JammyDodger is right. Which made me also look again in my messages (on the site where I now, freshly today, a moderator) and I see a Moderators group in there… and there’s the message—with an unread dot—from the test user I sent earlier today.
So, it didn’t send me a notification about an unread message in the Moderators inbox, when I was granted moderator powers. That seems like the correct behavior.
. . . which leaves me with the other point way above: Is it wise that it’s possible to have a site with no one in the Moderators group. Unless, I the Admin, think to check in the Moderators group . . . messages to moderators would never be noticed on my site. (Well, I’m leaving myself in mods for now for sure. 