if the waterpump beeder screw breaks off...

Damn thing! If it the head breaks off can you bleed the system [with the engine off, of course] by loosening the larger nut that the bleeder screws into?

Sure-- but do it while engine is running to do it right....

Sure-- but do it while engine is running to do it right....

I was told by a mechanic NEVER open the bleeder with the engine running

I was told by a mechanic NEVER open the bleeder with the engine running Opening is how it is bled, til a steady stream flows.

I was told by a mechanic NEVER open the bleeder with the engine running
You are either dealing with an idiot mechanic, or you misunderstood something he told you.

You never open the radiator cap while the engine is hot, however you can open while running and cool.

On the other hand the bleeder is to be opened when the engine is hot and running to properly bleed all the air out.

Damn thing! If it the head breaks off can you bleed the system [with the engine off, of course] by loosening the larger nut that the bleeder screws into?

you need to replace the bleeder if the part is broken . an engine with a bleed port valve is opened when the engine is running. you place a diaper under the area so as you open this valve it will absorb the coolant . you do this until no air comes out I would also have the pressure cap removed so the air bubbles out of the system . you do this / start this with the engine cold not hot ! this should remove most all of the air. continue to idle the engine until it gets up to normal temp pressure cap off .