Got it! Yaaay!!!
I had been intending to do a clean reinstall of OS X for several months because I was experiencing some wonky behavior, including a really weird Little Snitch glitch, so I figured that now was the time.
My first re-install of LS was no help because I had imported my old rules, but after it gave me some trouble I trashed the whole shebang and reinstalled again, and when the new installation asked me to connect to "rover" I clicked on "Forever," and that turned the trick.
Wondering whether I had set any "rover" related rules in the past, I booted into a pre-reinstall clone and found that despite the fact that Mail had always connected, I had never set a connect rule, nor did I find a deny rule that would have accounted for my issue... Got me beat!
Edit: So although I thought I had a Safari problem, I guess it was actually a Little Snitch corruption problem. Who'da thunk it?