HELP with Honda GX200 engine; I've tried everything I can think of!

Hello all,
I'm an Atlanta homeowner, not in the lawn/garden business, but I'm an owner of multiple big chainsaws and other small engines, 2 and 4 cycle. I've always maintained them on my own and learned over time (often the hard way!) how these things work.
However, this Honda is eating my lunch. It's attached to a CAT pump as part of an EXPRO2530 pressure washer that's been sitting out exposed to the elements for some time and was given to my wife and me to see if it would be of use.
I was able to get it started initially with a few pulls and was starting to work on the filthy front of our house, when it began to surge a bit and "backfire", for lack of a better term, and then died. I was able to restart it once and then it stopped and has not started again, or even coughed once when on full choke.
I'd already swapped what was in it with fresh gas, changed the engine oil with the proper type, added some pump oil to the pump, and changed out the spark plug (although the old plug looked good). No dice.
Since then I've torn it down, cleaned the gas tank, replaced the fuel filter, the fuel lines, the carburetor/all gaskets, cleaned the rust from the flywheel and any gunk from the cylinder fins, and replaced the entire ignition coil assembly and spark plug wire and boot. Good easy rotation of the flywheel, and good, strong spark.
Still no dice! Tried it with starter fluid as well, still not even a cough. Any help?
Thanks,
Truett Bridges

Thanks Roy; I'll check it out.

You might have experienced several problems initially since it sat outside and you would not know how well the unit was maintained by the previous owner. I would suggest that you check the flywheel key. It might be partially (or totally) sheared off- enough to throw off engine spark timing- it does not take very much. If that checks out, all that remains is some type of internal damage.

Try disconnecting  the oil sensor.