If you compression readings are correct then the saw has marginal secondary compression.
Have you done a leak down test on it.?
If it was used a lot there could be significant wear in the bore.
You said you replaced the piston & ring but did you measure the bore first ?
Not sure if this one had a blind barrel or not but I have sent back a lot of basket cases because the owners bought all the bits without first measuring things.
More than one had barrels so worn, the ring only touched in 2 or 3 places.
Single ring motors are not meant to be repaired.
You say you replaced the crank seals but did you pressure test them in both compression 5-10 PSI and vacuum 12" ?
You say the tank is not vented, well it has to be vented if air can not get in fuel can not get out .
Some have a vent in the cap while others have a vent in the tank itself and some tanks have a double vent to prevent over pressure.
Have you tried the engine with the kill wire removed from the magneto ?
Have you done an air gap evaluation of the magneto ?
How strong are the magnets in the flywheel ?
They should be strong enough to support the weight of the flywheel hanging from a length of pipe or under a steel shelf.
We have no idea about your mechanical aptitude
People tell us all sorts of things like "I cleaned the carb real good " = sprayed a full can of cleaner through the venturi
or
Replaced the seals - put them in backwards - failed to clean the shat properly sore tore up the lip
etc etc etc
And finally, a primer bulb on a hand held is actually a purge valve, it's job is to suck all of the air out of the carb so the diaphragm chamber is full of fuel.
As such, it sucks from the carb and blows into the tank.
It is not like a mower which really is a primer and floods the engine for starting.
If you have hooked one up to a non primer carb so it is pumping into the carb, it will not work as all you do is compress any air in the system.
Primers have a one way valve in them that prevents back flow and keeps a constant pressure head on the carb.