We turned the gas off at the sediment bulb under the gas tank on a 1922 Center Door. When we flip the handle to reopen the valve, it seems that no gas runs through the bulb or fuel line. There is plenty of gas in the tank.

I have a particularly cantankerous sediment bulb on a 1926 Roadster Pick up and refuse to use it to shut off the gas at that point except when absolutely necessary.

Do the valves on the sediment bulbs go bad, or are we likely dealing with a clog in the bulb/screen?

Ethanol blended gasoline is particularly problematic. I have struggled with problems similar to yours

I'm constantly blowing out the screen in my sediment bulb, part of me wishes there was one available with a little coarser mesh.

With a pan under the car open the sediment drain valve and see what comes out. If nothing at first run a wire up thru it to clear the crud. Slip a piece of tubing over the carb end of the line so it can be higher then the tank, with both valves open back flush the line to blow the screen out. Don't try and use too much air pressure, just enough to clear the line and screen. With the drain valve open it will help clear the sediment bulb.
Close the drain valve and check for flow. You might want to drain and clear the sediment bulb every so often.

If the handle turns and the shaft connected to the handle turns, the valve should work. Sometimes there can be a leak around the packing so that gas drips out, and other times there is a leak inside which allows gas to go to the carburetor even with the valve shut, but a clog is the only way to stop the flow of gas. There can be a clog anywhere in the line. If big enough particle it can block the opening in the tank and no gas will get to the sediment bulb or very little. If the bulb gets full it can also stop the flow, and of course the screen can get plugged. I am working on a car now which has been parked 26 years with gas in the tank. So far I have cleaned the carburetor, and replaced the sediment bulb and fuel line. I can get the engine to start with very rich mixture and choking but as I try to lean the mixture it quits and there is no gas at the carburetor bowl. I think the next move is to replace the gas tank.

There is no way of knowing the past of our cars unless we happen to be the very few who bought from an original owner or family of an original owner. Over the years rust and scale form in the tanks and old gas turns to varnish. So That could be your problem.
Norm

I once bought a truck that had been sitting for two years and it formed small lumps of coal the size of gravel within the gas tank. So, yes you may need to check the entire system.

x2 on checking the whole system. I found what looked like corn kernels in the entrance to the sediment bulb. I had to remove the sediment bulb from the tank to remove the kernels. I had to do this 2 or 3 times to get them all out.

I think the way I determined there was an issue upstream, was I removed the screen and with the cap off, I opened the valve, I had very little flow.

If you're using air to back flush the sediment bulb into the fuel tank, Take off the gas cap so you don't rupture the tank.

Jason, I didn't think the ethanol in our fuel could revert back to its original form like that.

Tin, I did not either. That is why I switched to non oxygenated fuel.

Good news. Temporarily solved. We ran a wire up the petcock on the bottom of the bulb, that dislodged something, and fuel is flowing again. We will investigate further, but the tips above proved helpful.