Boiler and controls cover does not cover cylinder, pump, pipes or radiators.
Funnily enough I have recently returned to British Gas for my rental property. I was previously with Adminicle, administered by Homeserve. Adminicle refused to come out last month because the heating was only partially not working. They also were year by year reducing the parts they covered. I only noticed after paying the last premium that they were no longer covering the pump.
My view for my home heating, is that I think British Gas will come out quicker if I am having to pay for their attendance, than if they can take as long as they want to turn up under contract. Perhaps a reasonable compromise may be to pay them a reduced monthly amount, but pay £50 per callout.