Per post #13, I had thoroughly re-cleaned the original carb. The first time I cleaned it, I did not remove the needle valve as you have to break the screw head off and replace the entire valve/screw/spring assembly. Well, this time I decided the needle valve was the only thing that wasn't cleaned twice, so it had to come out. I carefully removed the part on the side of the screw head that prevents it from turning and cleaned the needle valve and the port, along with every square mm of the entire carb, again.
Yesterday I had a very small window of light and good weather before night would again prevent me from working on the mower. So I removed the new carb and replaced it with the double-cleaned old carb. I also again removed the fuel tank, fuel hoses, and shut off valve. Opened the shut off valve, fuel flowed freely. I used this to drain the tank. Once drained, I removed the hoses/valve from the tank and used carb cleaner in both directions through the hoses/valve. No obstruction was found and carb cleaner flowed freely like the gas did. Since this model has the filter inside the tank, I couldn't do much with the filter. I hit it from both sides with carb cleaner and cleaner flowed without any obstruction. Also for good measure, I blew carb cleaner through the breather part of the air filter housing. Again, cleaner flowed freely in both directions. I found no dirt or obstructions anywhere. Time to put it all back together, it's starting to get dark.
I reassembled the tank, hoses/valve, and the old twice cleaned carb. I payed attention to the gaskets and now since they have been mounted for a while, they all held in place making it easier to reassemble. Always pay attention to the gasket orientation as those small holes are important. I put it all back together and held my breath. It started on the first pull !! Since I cleaned so many things again, I have no idea what may have been the culprit. If anything, I'd suspect the needle valve since this was the first cleaning for it. But why the new Honda carb didn't fix it is kinda puzzling. I didn't buy some Chinesium carb...I bought an OEM carb from a dealer.
I ran it for about 5 minutes and shut it off for about 5 minutes, then restarted. 1 pull. But restarts were always 1 pull. So for now I'm assuming it's fixed. I'll see how it goes when I actually get it out to cut grass next time. Here's hoping for 1 pull starts going forward.
Thank you ALL for the assistance and ideas. While I can't exactly pinpoint it to one thing, I know all of these things were good to do to ensure the engine is setup properly. I now know a little more than I did before and I'm glad I learned something. I'm nowhere near as knowledgeable as the pro's on the forum, but I am one step closer in a long mile or more to knowing what they know. Again, thanks all for helping me learn a little more about my mower.
Next I want to find an inexpensive means to measure the RPM's to make sure the engine is running at the right speed. It still seems just a tad slow on the RPM's.
Thanks
Dan