Why can anonymous users see my block, whose visibility is set to hide it for anonymous users?
The strangest thing fixed my issue.  I'm still not sure why.

I was placing the block in a certain region with the 'Context' module.  When I removed the block placement from that context and set it directly in the block's configuration page, the block stopped being visible to anonymous users.

Not sure why that would matter!  Placing blocks generated by views with the 'Context' module is not interfering with their visibility settings, but for some reason it was interfering with this one.  It's just a regular block with text, nothing funky.

In any case, that's what solved my problem...