Section 47 of the RTA makes no mention of driving to a booked appointment, or of insurance being invalid.
https://www.legislation.gov.uk/ukpga/1988/52/section/47I don't think the OP was asking that anyway, but the topic has certainly drifted off-topic quite a bit.
Those people driving a UK registered car without an MOT or UK Road Tax are breaking the law in Spain and probably think that the worst that will happen to them is that they'll get their car seized if they are stopped, plus maybe a small fine and the loss/gain of a few points on their licence. They probably figure that this is cheaper than spending money on a Spanish registered car, or putting their car through the re-registration process.
What they don't realise is that if they are involved in a serious or fatal accident (which isn't that difficult given the amount of pedestrians who look the wrong way before crossing the road, or the arrogance of some cyclists) then they will probably end-up in prison. If they've flouted the rules regarding vehicle safety and tax for a number of years then the courts will take a very dim view of their actions and want to impose a custodial sentence as an example to others.
Pete.