The waterproof grease needs to be added when the boat is out of the water, preferably on a warm day (although heat can be added with a torch) so that the grease works its way in to the mating surfaces between the rudder post and the gudgeon. I always did this whenever I hauled out: lots of grease and lots of working the rudder back and forth. If you add it when the boat is in the water (diving down), the grease will just float to the surface inside the shaft (waterline).
You must be greasing what I call the shaft log, the top where it penetrates the hull. I think technically the "Gudgeon" is at the bottom on your rudder and is a small area, and (I think) would not hold any grease. My skeg hung rudder had the equivalent to your Gudgeon and its up high, where a spade would have no other bearing than in the hull. Now that this thread came up, I need to look at how my rudder bearing is made, in a following sea the quadrant will be below the water line and whatever bearing is in there would let some water in especially if worn. Not to hijack but I'm pretty sure I have a hard plastic or teflon sleeve in the area you are greasing on your rudder and again the Gudgeon does not get grease its completely underwater and usually a pretty small surface and surely if you got grease in there it wouldn't last, so basically a wet Bronze to stainless surface. I see it loosening before seizing or galling, but I have never seen either one so, I defer to someone who knows more.