I haven`t tried it without the `2nd` pump. I can easily do so though. Hmmm, I`ll try that. I understand your theory, but I think the heat exchanger may be too flow restrictive for a gravity circulation ?
Yes, if its restricting the flow a pump would be a good idea!
I can`t do as you have in your second drawing. The heat store cylinder above the fire is vented to atmosphere. My DHW cylinder, UFH and system boiler is all pressurised. Your drawing doesn`t separate the 2.
That explains why its connected to the mains and why you need an exchanger

I'm not a fan of pressurised systems, but they have their uses!