How to improve band brake?
In the end I improved this brake's operation manyfold, but its still pretty mediocre.
While working on it, I found the brake pad (strip?) was slick.  So I presume some idiot applied oil to the brake in an attempt to make it quieter.

The main fix was to extract the band, and use a hot air blower (a workshop version of a hairdryer) to warm the band and I could see the oil running out of the band and dripping off.
I used a non-contact IR thermometer to make sure the top of the pad didn't exceed 150 degrees C and kept the hot air moving to not overheat one spot.

After a while the oil stopped dripping, so I had to use a blotting method; I'd heat the brake pad and then quickly wrap it in a couple of paper towels.  The oil was clearly visible wetting the paper.   It helped to massage the brake pad by hand, so I wore some decent leather gloves as heat protection.There were at least a dozen head-blott-massage cycles, and I stopped when the paper towel did not get wet.


On reassembly, the brake was okay when free-spinning in a stand.  There was a lot of fiddling back and forth with the cable tension and the two standoff screws that set the band's resting position, close enough but not so close that it rubs on the drum.
However when the bike is loaded, it absolutely screeches when braking.  This is what makes me think an earlier owner oiled the brake, which certainly quietened it, but at the expense of function.
I did lightly sand/scour the drum and the inside of the brake pad, and it really made little noticeable difference.

A better solution would be as Bruce suggests, and fit a rim brake caliper.  While this frame has a brake bridge, the rim lacks a braking track.  Given this bike is literally a shopper / dunger / beater, I'm loathe to throw more money at it.   However I did replace the fork to get canti mounts, and the front wheel has functional and effective V brakes on it now.