So, what happens if the Bus card-reader is broken, and the driver
waves you on - happens a couple of times per month on the routes that
I use (308, 678). Take it I should insist on paying cash?

You can add the 8, 277 & 15 to that list and countless more probably.

The official TfL position is that you should pay the cash fare and request
a refund of the price difference from them...

Unless you were capped that day in which case you would be entitled to
a full refund.

Or if that journey was the last one that day and took you past a cap then
you would get a refund of any extra you paid in total beyond the cap.

Not a very elegant solution.