There's an old trick that, Sometimes, worked on a saw that had been left sitting with fuel in it,
If the carb is mostly gummed up with the oil left after the gasoline has evaporated.
Make sure the saw has fuel and can get through the line to the carb, no clogged fuel filter or bad gas line.
Next Put your thumb tightly over the carb bore and totally block it off. You want NO air to get around your thumb.
Now pull the recoil about a 12~15 times.
This will purge the carb of oily mess by pulling a LOT of gasoline into and through it.
it will NOT remove debris or "fix" a clogged screen or passage clogged with trash.
Now the spark plug will get fouled/wet from this & the saw will get flooded
Get a fresh plug and and be ready to pull it and clean the gunk off and
try to start the saw a couple ~3 times as the gunk from the carb (& fuel line) will foul the plug a couple of times, after this.
This used to work, Sometimes, back in the days of non-corn gasoline.
Worth a shot IF you feel like fooling with it.
Just Don't jerk and abuse the recoil. Pull it in a normal fashion...always.
But I would still figure the carb for a rebuild, meaning a new kit, not just wiping the parts off and throwing the old diaphragms back in it.