Check your pump speed. One I looked at with similar symptoms had had people trying to improve the balancing and muttering about sludge, but the pump was on speed 1 out of 3. Speed 2 fixed it with one click. Speed 1 must always have been low, for the 3 bed house. You may well need speed 3.
Otherwise, you could do the old thing, which is to take out the offending rads to the garden and flush them with a hose - turn them upside down and back a few times when half full and keep at it until the water runs clear. You need a helper, but it's cheap to do

. (Pot/tube of jointing compound for the rad valves.)
While each rad's off you can turn the Htg on (CH only, but boiler stat OFF) with only that radiator's flow valve open, into a bucket. It's not a power flush but it's a lot more flow than it's used to and you'll be checking it's ok. If it's got 3/4" rad valve fittings, a garden hose connector or washing machine fill hose will fit, going to a drain.
If there's no flow, connect the outside tap to the drain end, (not open much, to begin with). Disconnect at the outside tap end every few seconds, and let it go to drain, to make sure you have water going up and down the pipe ok.
Be mindful of the level of water in the header tank, and check its overflow will warn you if it's getting full.
It'll take half a day but maybe save you £4k. That's nearly as much as
I earn

.