That's that then! From the symptoms, it sounds like an On/Off (not modulating) control-stat on the boiler. I'd expect it to cycle On/Off on DHW (but not necessarily rapidly) as the heat exchanger in the cylinder cannot transfer the boiler heat output. On CH, often the rads dissipate the heat so well that control-stat point is not reached before the roomstat is satisfied.
I recently changed my pump as it was noisy and vibrating. I hoped to just change the motor and impeller section, to avoid disturbing the pipework, but when I removed that there was a stainless steel ring pressed into the casting, near the eye of the impeller, so I had to change the lot.
But when I examined the old pump I was surprised by how much clag was in and behind the impeller. I was surprised it pumped at all, but now I've cleaned it up I'm sure it would be as good as the new one, so I've kept it in reserve. Before buying a new pump it might be worth removing yours (quite straightforward, just 4 small (M5 I think) socket head bolts) and give it a clean.
