While this isnt a water based system, I thought it might be a good place to receive guidance on this specific part.

Changed the filter, reopened the valve but the oil flow has stopped.

Ive since added oil to the tank so I think there should be enough pressure to restart the flow.

I'd like to determine whether this valve, which I am referring to as a "Firomatic" valve, is broken and stuck in the closed position. Not sure how to go about it.

Any guidance would be appreciated.

 

These valves are supposed to be open unless there is a fire and the handle melts and the valve closes. The stem is pulled up by the hand wheel and many threads will be showing as in your picture. When closed the threads almost disappear. I have removed the handle and gently pulled and released the stem to verify it was moving until I decided to believe what I saw. It is spring loaded closed.
According to your pictures I would assume it is opening.

 

I left the handle on, towards the top, and used it to pull the stem.

It has some movement and has a springy feel.

Considering the worst case scenario, I think I now need to determine whether the stem is still connected to the part/piece which actually stops or allows the oil flow.

 

L,
From your pics you have a 2 pipe system for your oil delivery with a CHECK VALVE on the SUPPLY SIDE so the oil doesn't backflow into the Supply and after going through the pump is returned back to the tank by the Return line with the other valve on it which must be open when running the burner.

The oil does not flow by gravity because of the check valve for one. Where is your tank located. Is it inground. Oil does not flow uphill.

As far as bleeding goes, 2 pipe systems are self bleeding when you turn on the burner. It may take a couple of cycles or you can jump out the motor and bypass the safety until oil comes if it gets difficult to pull.

The CHECK VALVE on the Supply only opens when the burner runs. The force of the pump opens the gate on the check valve and allows oil to flow.

The Firomatic valve is open when the wheel pulls up the stem and the threads are visible and the wheel itself has a lead insert which melts at 165 deg. to close the valve in case of fite.

Hope this helps a little.

 

If you can purge the fuel pump and get oil out at a decent flow the valve is open. The valve should look like your middle picture to be open.