If it is an old open vented system with a feed and expansion tank as you describe, it is almost certain to hold a large amount of sludge and sediment which will cause partial (at least) blockages.
I suggest you go to the DIY shed tomorrow and buy two litres of Sentinel X5
400 chemical cleaner (not any cheap substitute they may offer, and not an acid cleaner, which would be effective but needs more skill and is more likely to cause leaks in old radiators. Buy two litres of X100 as well but you will not use it for four weeks. The chemicals will cost about £16 a litre. Buy from Screwfix if you can.
Find the drain cock on your radiatior system.
Climb into the loft and tie up the ball valve in the SMALL tank. Bale all the dirty water and mud out of it and sponge it clean with a bit of bleach. If you find a blanket of fungus, or white bacterial jelly, you will have an extra step later.
Ask your assistant to drawe two buckets of water out of the drain cock. Tip the X400 into the F&E tank and untie the float so the tank refills. Adjust the float so the depth is only two inches above the feed pipe at the bottom.
Now turn the boiler on. There may be some bubbles you need to bleed out of the rads. Turn the pump down to speed 1 and see if this stops water gushing out of the vent which is the 22mm pipe bent into an inverted 'U' over the small tank. BTW how high is the top of the U above the tank?
You will know the cleaner is starting to work when the circulating water goes jet black with dislodged sediment. You can leave it working for four weeks with the heating running before you have to drain it out, rinse to clean, and refill with fresh. When you are sure you have finished cleaning and draining, add the X100 to prevent future corrosuon, tying up the float and draing out a couple ofbuckets first, so it will be drawn down when you refill.
As for downstairs radiators being cold, turn off all the hot ones and I expect the cold ones will then heat up. We can talk about rebalancing another time.
My finger is tired now.