Sounds like the check valve sticking shut was the main problem and it could happen again. Do you know if fuel containing alcohol was used, that's the main cause of these things going funny? If you feel like experimenting, just remove it and not bother replacing it, as I mentioned it's only there to enable the purge bulb to work and the saw should run fine without it. Not so with the main jet check valve, it is essential. To remove it, the best way is to tap and thread it for a 4-40 machine screw and use a screw threaded into it to pull it out. Do NOT try to pull it out with the tap bit, it's brittle and if you break it off in the valve you will be in more trouble, just be aware that removing it will destroy it and the purge function will no longer work.
When pressure testing the carb by blowing into the fuel inlet port, it SHOULD hold pressure up to 7-10 psi if the inlet valve is seating properly.