I finally took the car in for an MoT.
Unsurprisingly, it failed. The handbrake efficiency was below the requirements and there were a few areas of corrosion so not bad for a 42 year old car. I've cleaned, adjusted and greased the rear brake shoes and the handbrake is better. The garage is going to do the welding so that will take care of everything and it will have a fresh MoT (and an empty bank account).