Look up "mains flush" here. You can do quite a lot of good by blasting out whatever will move. To be effective you need to work out your heating system (draw it, with the valve(s)) and shut things so you're forcing water through one rad at a time and not having it short-circuited by the boiler or cylinder. Don't forget to deal with the f&e tank, and vent.
Fernox DS40 and Kamco FX2 will dissolve it, but need neutralising afterwards. They find weeps.
X400 is gentle, other Fernox chems and Sentinel X800 are in between. There are several others.
One of the last boilers I took out had been powerflushed ( a one day not too aggressive sort of job) by BOTH British Gas and a Powerflushing Company (red elephant one, normally ok) but the pipes from the boiler were still only about 8mm diameter clear, inside the 22mm copper.
If 'twer mine I'd mains flush it, put X400 in for a few weeks then repeat the flush, then maybe fit a filter, depending on the boiler type/age. Big/old/iron are less fussy. Combis much more.
Unless it was really clean I wouldn't put inhibitor in - it sticks to suspended particles of oxide and makes a slimy mess.
Your flakes can stack up and block things, but "grubby" liquid? You could have Guinness in there, as long as you've got enough flow, it does the job, in an oldie.