Re: How many strokes of the....
Nivek, the difference between your tip an mine is that mine is affixed to a solid, slightly crooked wand. I have several different guns that have the same setup. I also have one with a flexible wand but I never even looked at that one because I figured that it would be difficult to get that flexible hose to snap on to the zerk fitting and it does not have marine grease in it.
On my first attempt to grease that swivel fitting, I did not get any semblance of a 'pop' of the tip onto the zerk. I still gave it one pump. There was no resistance so I looked up inside, with an LED penlight and grease had splayed all around the fitting. I brought the gun inside and popped it on to a spare fitting. anchored in a vise, just to be sure that a zerk could lock into the tip. It did, with clarity.
I then went back out and tried it again. I still could not get that same 'pop' but the tip did seem to grab the zerk. That is when I gave it 4 pumps with seemingly a bit of resistance. Afterwards I looked back up the chute again and there was still a fair amount of grease around the fitting. I don't know if that grease was there from my first attempt or my subsequent 4 pump. I think I will go out a play with it again.
I would go to your local discount autoparts store and get fixed wand grease gun before I enlarged the access hole. They are very cheap. Here is an example. It pretty clearly shows a narrower tip than what you have.
Lever Action Grease Gun
Furthermore, I am not sure what the effects are of having water rammed into that cavity during operation. I just can envision that it would be good to have water forced into that swivel area. Who knows, maybe when you are underway, that hole is above the water line. I have not bothered to make a determination. However, it seems that I have seen some kind of parts breakdown, for some variation of the Stringer, that showed a plug for that hole. Mine, circa 1979, had no provisions for any kind of plug. I would have to go double check but I think I used a 1/2" npt pipe tap, that nicely threaded the rather large bearing surface of that hole. I then went to the local ACE hardware and got a nylon pipe plug that used a hex key to install.