My parts vendor asked me to bring the pump to his shop. Said he'd give me another new pump if we couldn't determine how to correct the problem.
We replaced the brass fitting with a modern zerk fitting. We unscrewed the packing nut entirely. Low and behold, we were able to slowly add grease and could see grease beginning to emerge between the shaft and the packing. We concluded that the packing nut would seal the greased area even if just a couple of threads were engaged.
So, my bad for not understanding that "the packing nut must be loose" actually means that "the packing nut must be fully backed off the threads". Sorry to drag so many of you through what turned out to be operator error, but thanks to all of you who tried to help.