You are absolutely correct I was just about to post this for anyone else looking;
I went out to remove the shaft but decided to search one more time. I found the fitting deep in there covered in crud.
I went to NAPA and bought a Lincoln model 5803 Needle Nozzle fitting for my grease gun ($10.99) It fits perfectly.
It took 14 pumps on my mini grease gun handle before seeing grease on the far side; so I imagine it was fairly dry...
Needle nozzle pointing at fitting: