OK, I have done some trouble shooting, and hopefully it helps

The toilet bolw sits 1/2 full, if I flush, it burps, fails to flush, but fills properly.
If I then flush it again immediately, it flushes properley.

If I take a 5 gallon bucket of watter, and dump it in the bowl, it flushes quickly

I can not hear the toilet leak, or fill, during the night and the level in the reservoir stays full.

If flushed with a full bowl, it flushes well enough.

Personaly I suspect a leak, between the reservoir and the bowl, permitting air into the syttem, breaking the syphon, However I have not seen mention of this anywhere else, and does not sound like a common problem. I would like to hear from others before I start taking things apart.

The toilet bolw sits 1/2 full, if I flush, it burps, fails to flush, but fills properly. If the bowl fills after flushing and then drains away then the flapper's doing its job OK.

Personaly I suspect a leak, between the reservoir and the bowl, permitting air into the syttem, breaking the syphon, If that were so you'd have more water in the bowl not less and the flapper has nothing to do with the siphon action. When you have a full bowl when you go to bed does it lose water overnight? Back to you, Tom

Try looking at the flapper when you flush your toilet. Flappers are designed differently to allow enough water to leave the tank before it closes. Putting food coloring in the tank and then finding color in the bowl shows the flapper is not sealing properly and needs to be replaced. A flapper that closes to fast needs to be replaced too. Make sure you get the correct flapper for your toilet. Check it out.

A flapper that closes to fast needs to be replaced I disagree! The flapper needs the linkage adjusted not replaced, Besides, If you go back and read the complaint you'll see the bowl fills.

Putting food coloring in the tank and then finding color in the bowl shows the flapper is not sealing That's correct but that would put water in the bowl while the complaint was the opposite. Back to Plumbing 101 for you, Cheers, Tom,

After many weeks trouble shooting, and guessing, and asking for help, I am 90% sure I have solved it. I will leave it here, as a "quiz"

Tomorrow I will start giving hints. I got this sucker figured now.

No guesses? Here is a hint. Works fine when the cover is off the resivoir.

here is a hint. Works fine when the cover is off the resivoir.
Well then. Problem solved! You say,
The toilet bolw sits 1/2 full, if I flush, it burps, fails to flush, but fills properly.If it works with the tank lid off then you came up with your own solution and don't need our help. I've never did a toilet repair by simply removing the lid but I'm always looking for new solutions. Good luck, Tom

The fill hose, that goes to the fill tube, was being kinked, when the reservoir lid was being placed on the tank.