I used a piece of 1/2" NPT pipe on an Echo 4600 and it gave me a lip I could put a gear puller on.. worked well
Could it be the throttle linkage is too long, causing it not to rest on the stop screw? I'd firstly make sure that your carb gaskets, etc are good before digging in deeper.
Since it needed a new P&C, the problem could be a case gasket that caused it to run lean and burnt out the original..
As for the compression tester, it doesn't take much of a leak around the threads to really lower the pressure.. many compression testers have an adapter that screws on with longer threads.. perhaps this adapter is still on yours? check to see if yours still has it on!