The key takeaway is to remember that the bootstrap is on the clade, not the branch -- but the standard convention is to show the values adjacent to a branchpoint, which fueled my mental fog.
Mentally now, I'm envisioning the tree more as a series of brackets enclosing clades, with the adjacent bootstrap value assigned to that bracket
An alternative mental model I've cooked up is concentric rings -- with the bootstrap values assigned to each box being the outermost value within that box.
Obviously each of these is non-canonical and I don't plan to include them in a manuscript or slide deck, but they've helped me picture what is going on and remember it.
I also was reminded by commenters of the importance and challenge of stating your problem correctly when you want help. I had shown a tree, which was really hacked off a larger tree, without explicitly stating it was a rooted tree. DUH! Part of that is discussing rooting trees can give me significant anxiety, because I'm never quite convinced I can declare something a kosher outgroup in my trees.
One final note: I'm not the only person with issues around bootstrap values. A recent bioXriv pre-print describes issues in the Dendroscope viewer which can arise after re-rooting trees.; I use Dendroscope and generally like it, but it is a warning that