i'm no expert but with my open vented system, i have everything turned off, then bleed all the rads. Bleed all the rads once, then re-visit after the last rad is bled just to make sure.
Then I swtich the CH on and let it run for some time( if any TRVs are fixed, turn these up to max). . IF it's a new system ,try leaving it running for a while?
I also have a CH pump in my system boiler - with everything off (system is cold) there's a screw in the middle that can be turned to release any air trapped in the system. Turn this until a little water dribbles, then tighten the screw. Not sure what type of CH system you have
Again, i'm only sharing what's worked for me in the past from seeing advice on other forums like these - and re-iterate, i am no plumber!
Since my previous post, I've also taken a look at the expansion tank and the water looks really grimey, but not smelly - so I think the whole system needs draining and flushing for a start.
I've also spoken to another plumber who's been recommended and he didn't think that the pump was to blame.
So, with my absolute lack of knowledge and not wanting to make matters even worse, I've called in this other plumber and I'll see what he tells me because he told me that the water in the expansion tank should look reasonably clean.
If the feed to the system from the bottom of the header tank is on the inlet side of the pump, and the expansion leads somewhere off the flow side, then water will constantly be overflowing the expansion pipe into the tank. If the feed and return are reversed (as in your case?) then air will be constantly drawn down the expansion and into the system.
The inlet and expansion must be sited close together, no more than a foot apart otherwise problems will occur.
as well as the other problems one you should consider is when you bleed the rad hold a lit match or lighter to it. if the air lights then it's hydrogen and the problem is the system or radiator itself making it. the rad or the system would need a clean out to solve it.
if it's the system drawing in air then it won't light.