I am not a pro, just an experienced DIYer.
You have already changed the pump and it didn't help. You have evidence of black in the radiators. I would get on with giving it a simple clean which will take a few hours of your time and cost you about £15. Very likely it will give you a cure, if not, it will do nothing but good in getting some of the sediment out, and will have cost you little.
Go into the loft, tie up the ball valve and bale out the mud from the F&E and sponge it clean (you don't want to add to the sediment load)
drain a couple of buckets of water out of the rad drain cock
Buy a litre of Sentinel X400 from any DIY shed or plumbers merchant for £15. Untie the ball valve and stir well to mix the cleaner with the incoming water so that it flows down to replace the water you drained out.
Run the system as normal for 4 weeks and it will loosen the sediment. You will know it is working when the circulating water goes inky black.
In the first few days, turn up the pump to fast (watch the F&E to make sure it is not pumping over) and go round turning one or two of the rads full on at both ends while closing the others, to increase the flow through those few rads, then do the others a couple at a time in the same way, and so on, then set it back to normal but with all rads open. If you find any of the rads are cold, turn down or close the hot ones until the cold ones heat up so that you know the chemical cleaner is circulating through them
After 4 weeks, turn off the boiler and tie up the ball valve, and drain out all the water with its loosened sediment, and refill/rinse/drain until it seems fairly clean. On final fill add a corrosion inhibitor such as Sentinel X100 for another £15, and balance the rads as shown in FAQs.
If you can do basic plumbing and can afford £100, add a Magnaclean. I guarantee you will be amazed and delighted by how much sediment it collects, and it continues working permanently to prevent black sediment collecting into a new blockage.