Is the sediment on both hot and cold water lines? Or only on the hot water line? You said that you've had to clean out the hot valve on the bath tub. Maybe you should run a test to see if there is any sediment in the cold water line. If the sediment is limited to the hot water line, I would suspect your water heater. I recall there was a problem some years back with the filler tube in conventional water heaters, whereby the combined effect of heat and time caused the tube to decompose. I don't know the specifics other than sediment was coming out of the hot water line. You describe the particles as black, white and gray, which could indicate a similar problem. I don't think it would be the resin in a water softener, because the resin is brown. Don't put a sediment filter on the hot water line, because it won't take the heat.