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Your engineer quite correctly has highlighted the common faults that cause a boiler to lock out ( ie over heating) namely a faulty limit stat and failing that a lack of ciculation through the boiler hence a replacement pump.
By flushing out the system you remove the sludge and any other debris that could be reducing the flow of water.
Similar results to power flushing can be achieved using a heavy duty cleaning agent and flushing out without a power flusher but it does take longer.
£600 to £800 for power flushing is not unusual.
A cleaning additive cost around £20 and you leave it in the syatem for a period of time . Then flush the system through with a hose etc..
Hope this helps
Having changed the component that controls overheat lock out, it narrows it down.By lower the stat on the boiler ( as opposed to the room stat )will reduce the likelyhood of over heating. If lock occurs having reduced the room stat the system temperature will still remain the same but will stop calling heat sooner and perhaps shuts off the pump too soon causing lock out.
I think flushing out is a good course of action and as to whether you decide on power flushing is down to you ( two schools of thought about power flushing!)
Glad to help and always grateful for a bonus!
I.m in the cotswolds, born and bred.
Not too far from me , Chalford .