The carburetor adjustment screws are very finicky.1/8 of a turn can make it not run properly.
The question is does it have spark AFTER IT QUITS AND BEFORE IT COOLS OFF.
I cheat by using starting fluid . If it doesn’t start on that stuff it probably doesn’t have spark.
I have spark testers also . When using starting fluid don’t try to use starting fluid continuously to try to keep it running because the rings won’t be getting any oil.
I looked at all the things you replaced. go on you-tube and watch several videos on installing new gas line .It is easy to get that wrong especially if the last guy that owned it did it wrong
I put a new carb on an echo edger. It worked great until I tried to start it again. I thought it was toast .I tried starting fluid and it started and ran great.
Looking again at your parts list.I don’t see any coil or major ignition parts.
I have an old echo that starts and runs great until it gets hot then it stops and won’t restart until it cools off . It doesn’t have spark after it gets hot. The weed trimmer is so old I can’t get a coil for it. Some day I will tear it apart and check the condenser.
New shortbloc arrived today, and I’ll tear into it with my 8yo son tonight 
Will pull the head once the swap is done. I’m realizing now this will be at least the second engine on this one…
At least the wife has a good sense of humor about my “craigslist projects”…
Teaching the kiddo about small engines is super valuable. My girls don’t really have any interest in mechanical things, but I teach them as much as they will suffer through. 
The transplant is complete and it’s running like a champ!
Pulled the cylinder head off the old engine and it’s obvious what caused the compression loss. Piston ring appeared to be intact, so not sure what caused the scoring on the piston. Since it was on the exhaust side, I wonder if something worked it’s way in past the muffler…
Thank you again to all who provided guidance and troubleshooting ideas. If you’re keeping score, ‘loss of compression’ was the winning answer.
Nice necrotopsy!
(and congrats on the surgery success!)
Womp womp womp. Last place trophy for me.
congratulations
I missed this thread. Just for future reference I have a small engine compression tester with a full set of adapters for various plug sizes. While I generally don’t like loaning tools, I’ve been known to make exceptions and could always meet someone at DMS once we’re open again.