Oh, it does sound like my house when I moved in. I have a C Plan, no question, it is a C Plan, but two pumps and two motorised valves, not tank thermostat for the central heating, and I have an immersion heater which is controlled by how much solar I am getting.
So C Plan means the domestic hot water (DHW) is heated using thermo-syphon not a pump, but the central heating still needs a pump. It seems it was a preferred method with oil boilers, as it allowed the boiler to cool by heating the DHW.
However in summer is causes a problem, as if you leave the boiler on, it fires on a regular basis, and so wastes energy, so the C Plan was altered to include a tank thermostat for summer use, this allows you to leave the boiler on, and ensure the water is hot enough to stop legionnaires. Personally I would use an immersion heater.
The C Plan was further modified to reduce the stored water temperature in the winter, with a motorised valve, but this was a bit pointless, as it removed the whole point of the boiler being able to cool by heating the DHW. However this means we have three basic wiring diagrams for the C Plan.



as the idea progressed.
So first the immersion heater, I had solar panels fitted, and a special unit to use excess solar to heat DHW, how it works at this point does not matter, but what is also does is give me a display to tell me how much energy is being used to heat DHW, and this is around a tenth of the energy used when using oil, so even when oil is cheaper to electric, using the immersion heater in summer is clearly the way to go.
I found the idea that turn off the pump, and the central heating will not work idea is flawed, once the flow is started, up-stairs radiators can continue to heat, since I don't use oil for DHW in the summer, this is no longer a problem. Also the whole idea of turning off a pump did not work with two zones, the zone with pump turned off just got a backwards flow.
I still have the Nest Gen 3 connected up, so I can in an emergancy use the oil for DHW, it also means if the Drayton Wiser thermostat fails, (discharged battery for example) I have a back-up.
But I never set it up to heat DHW with the oil boiler, but it does heat the DHW as can't turn the function off, however, it seems likely you have a S Plan, but can't be 100% sure without being there working out which pipes get warm when.
For a non modulating oil boiler, the Y Plan is likely the best, it allows individual control, but default is DHW, so boiler can cool by heating DHW, once we move to modulating boilers, be it oil or gas, the whole idea of how to control changes, and the TRV (thermostatic radiator valve) is King.
The problem I found as an electrician, was to work out what I am working with, the boilers where the pump etc, was built in the boiler, made it so much easier. The heating engineer may know which boilers need a delay to cool down, as an electrician I don't have a clue, the reverse with plumber, he is also like the electrician only knows half of the system, so as a pair we could work it out, but for someone to know the whole system, we need a heating engineer, and an engineer means someone with a degree, so they want paying for their skill.
I was lucky, I knew the electrical side, so only needed a plumber, I thought, but then he called in the heating engineer, and between the three of us, we got it sorted.
It is not perfect, most central heating works on the near enough engineering idea, to make it perfect costs too much, so you want someone with experience, so he knows how far to go, but not someone with the attitude I have always done it this way, so will continue doing it this way. Central heating has evolved, and what was done 50 years ago may not work today.
My parents central heating was used to warm the house before they got up and lit the fire, it did not run all day, we tend to forget that. Where I volunteer we still have central heating which uses a boiler, i.e. steam not water, and a one pipe system, where the steam is vented to atmosphere at the last carriage. Really cold carriages if diesel hauled, as no steam for the heaters.
We simply forget such systems ever existed, but oil boilers last a lot longer than gas ones, so with oil we see some antiquated systems.