Update:

I had to change a bunch of stuff around, so I'm not sure if I've really solved the problem, but I did get it running. Here's what I did to it:

Took it out of the plane and installed it on the test stand again. I installed it so that it was upright rather than inverted.

Put on a brand new fuel tank with no fittings anywhere in the lines. Used all brand new tubing.

Removed the static pressure line from the carb.

Elevated the fuel tank to try and give it more pressure at the carb.

I started out spraying it again with the starting fluid. As before, it would start and run for a few seconds. I did that about 4 or 5 times, hoping that one of those times it would keep running. It didn't. Since that approach wasn't working, I figured I'd try something different.

I finally got so mad that I figured I'd see who gave out first, my arm, me in the heat, or the motor. I told my wife that if she looked out the window and saw me laying face down in the driveway, to just call 911 and take me to the nearest hospital. Then, as was suggested, I opened the throttle all the way and closed the choke. I flipped it until I heard it pop. (Took about 25-30 flips). I then closed the throttle, opened the choke, and after a few more flips, it started and ran. I ran it for about a minute, blipped the throttle a couple of times and shut it down. I didn't want to overheat it.

Now I have to figure out a systematic way of undoing all the different things I did, to see which one was the problem. Maybe none of them, I'm really not sure. Hopefully, now that it's run, I can get it started in the plane. I want to put the old tank and lines on there first, to see if there was a problem with any of that. Then maybe turn the motor inverted and see if it starts. Then I guess I'll put it back in the plane and see if I can get it started. Oh yeah, and I have to reinstall that static pressure line at some point. I don't believe that had anything to do with it, but who knows at this point.

arobatx, the lines were correct as installed. For filling, I simply use a "T" in the line from the tank to the carb and cap the fill line with a plug. I tested the fuel system and found no pinholes or leaks anywhere.

It almost seemed as if the carb was air bound or something. I'm hoping now that I've had the engine running, it will be easy to start. I'll keep you posted of further developments.