Sounds like you're on the right track, if you had black goo in the filters you can bet the tank strainer is clogged, if it's still there. Sounds like you have a small compressor, you could probably pump it up until it shuts off and then transport it to the machine, should have enough in the tank to blow out a fuel line.
As far as the priming pump, the Deeres I remember had a diaphragm pump with a little priming lever on the side, when you move the lever it's actually pushing a spring back and the spring pumps fuel on the off stroke. When fuel pressure comes up it holds the diaphragm up off the cam and the lever gets real easy to move because it's no longer doing any work. If the engine stops in the wrong spot and the fuel pump lever is up on the cam lobe, you can't move it enough to make it pump, have to bump the engine over a bit first.
Another thing to check is the fuel outlet fitting on the injection pump and the return line to the tank, make sure they're clear. A clogged return will make the engine lose power and/or die also.
As far as the priming pump, the Deeres I remember had a diaphragm pump with a little priming lever on the side, when you move the lever it's actually pushing a spring back and the spring pumps fuel on the off stroke. When fuel pressure comes up it holds the diaphragm up off the cam and the lever gets real easy to move because it's no longer doing any work. If the engine stops in the wrong spot and the fuel pump lever is up on the cam lobe, you can't move it enough to make it pump, have to bump the engine over a bit first.
Another thing to check is the fuel outlet fitting on the injection pump and the return line to the tank, make sure they're clear. A clogged return will make the engine lose power and/or die also.