Too me it sounds like when the engine first starts it's a little flooded. Once it burns off some of the excess fuel it leans out and starts to run well, but then runs out of fuel and stops again. The reasons for it being flooded could be from excessive priming, or from having the fuel tank too high so that fuel is slowly siphoning into the engine. I don't think the needle valve is open too far or it wouldn't lean out and stop...unless you're not drawing fuel at all.
For testing purposes I would put the top of the tank even with or just slightly above the needle valve. That way you don't have fuel running in and flooding the engine, but you also don't have fuel running back out to the tank. In that situation, if you get it running and it still speeds up and then quits, then I would think you should open the needle valve even more, although to be honest 4 turns seems like a lot to me.
Better yet, leave the tank disconnected. Prime the engine and try to start it. You should be able to get it to start and run off the prime for a couple of seconds. Once you've figured out to get it to start off the prime consistently, then you're ready to connect the fuel line and try again. If you have the same trouble, then try opening up the needle valve another half turn. If it gets worse go the other direction.
Regarding the electric starter - I feel your pain. I have a 1/2 a starter that I use sometimes. Be extra careful the engine is not flooded when using one since you could bend the rod if it is. Also, as mentioned previously, the engine wasn't designed with those forces in mind, so it may wear prematurely in places you wouldn't expect.
Take all of this with a grain of salt as I have no experience with that engine in particular. I do have several Cox and Norvel/AME engines however.
Good Luck - your persistence reminds me of when I was a kid trying to start some of my engines.