Honda Mini Tiller stops at full throttle.

I recently borrowed a Honda Mini-Tiller with a Honda GX31 engine from a friend that is driving me crazy. It hadn’t been run for over a year. The bulb on the carburetor was dried and broken and my friend told me that was probably the only problem with it. When I got it home, I pulled and disassembled the carburetor and found that the first membrane had a small crack in it. So I ordered a new carb. I also pulled the spark plug and observed a very nice spark when I pulled the starter cord. I drained the oil (it was very dirty) and cleaned up the engine till it looked almost like new then refilled the oil with 10W30. I also drained the fuel, checked the filter (I could blow and suck air through it easily) and refilled the tank with premium fuel. I pulled off the valve cover and noticed that the gaps seemed to be larger than I would have expected from my previous experiences with rebuilding small engines. But I left them alone since the unit had worked ok before it sat unused for a year. When the carb arrived, I installed it that evening. I was very disappointed when it wouldn’t start that night. I went to bed thinking about what I would try next. But the next morning I pulled the cord with the choke on and it kicked a little. I turned off the choke and it started on the second pull. Why wouldn’t it start the night before?
Smoke came out of the exhaust for a couple of minutes when I started it that morning, but it ran well and worked good for about an hour. Smoke occasionally came from the exhaust as I worked, but not excessively. Then suddenly it simply stopped. It restarted with one pull but it would only run at working speed for a few seconds before stopping again. It idled fine. So I pulled the tines out of the dirt and tried a few things. I learned that when I pulled the throttle handle all the way, the engine would rev up quickly but die when it reached full speed. It would still start with one pull (no choke). If I slowly increased the speed, it would run great until it reached about 80% speed. But when I pulled the throttle the last little bit, the engine increased speed then stopped. If I quickly released the throttle as it was stopping, it would continue to run at idle speed and I could again rev it to 80%. Thinking it had simply overheated (I worked it hard while it was running well!), I let it cool for more than an hour. But when I returned, it again started on one pull, smoked a little, but stopped when it reached full speed. I got a little more work done by running it at about 80% speed, tilling for several seconds, then releasing the throttle as soon as I heard it start to stop. But after fifteen minutes of this, it wouldn’t start again.
I took the unit to my garage to dig into it again. I removed the valve cover and set up the gaps to spec. But it still wouldn’t start. So I checked the spark and noticed it seemed to be weak and intermittent. (Remember it had been very strong the day before!) I bought and installed a new spark plug and it started on the first pull and idled well. So I took it out to finish the job. Again it ran well for about an hour then started having the same symptoms of stopping when it reached full speed. I removed the spark plug and cleaned it with a wire brush. This seemed to help a little but only for a few minutes. Cleaning it a second time did not help. I tried to run it at 80% and play with the throttle handle, but gave up after 5 minutes.
Now I don’t know what to do next. Has anyone experienced this with a small engine? Could the piston and rings be damaged or worn? This could explain the smoke but not the other symptoms. Could the new spark plug already be fouled? Could the carburetor need to be cleaned or replaced again? Any ideas will be greatly appreciated.
Thanks,
Greg