Ugh, only minutes after typing up that question, I fiddle some more and find the answer. Apparently Visio is very picky about what the connector sticks to. I was casually dragging the connector and linking it to the ClassName
bit, like so:
Instead, you should fiddle with your mouse, until you see this:
The easiest way I found so far to do this is moving your mouse to the left edge of the Class box until the green line surrounds the entire thing.
Leaving this up here, hoping it may help someone else too.