The fact that you have not got very far, suggests to me the root cause of the problem is elsewhere.
Yes! it is essential, radiators need to be balanced so each radiator gets sufficient flow to give the desired temperature for the room.
The problem radiator works ok when others are off, but not when others are on.
This is caused by too much resistance in the pipe work.
Usually there is a by pass valve which allows water to go from the 'flow' to boiler 'return'. Now if this valve is open too much, the water will take the easier path and go back to boiler rather than go to the problem radiator.
The radiators subject to this type of problem are those furthest away from pump.
Also if the heating system is divided into two sections all the lockshields on one section may have to be opened/closed a little for the other section to perform.
The flow through the radiator is dependent on water being pushed against what ever resistance exists. Which brings me to another important issue, the pump.
A friend had a similar problem, radiator did not heat up. Radiator removed, cleaned and replaced. made no difference(yet other radiator in same room hot).
Finally located by pass valve and closed it down and radiator got warm but not hot.
Next removed pump for examination, black scale formed around inlet and outlet. 20mm orifaces down to about 7mm so pump could not push water to problem radiator.
Decision made to replace pump and isolating valves.
New pump fitted, readjust by pass and rebalance problem radiator, no further problems