The engine running for 5 or so extra seconds means that the keyswitch/kill circuit isn't working, as it is shut down by the fuel solenoid. So your problem may be related to that problem {the keyswitch area}
Take a pic of your engine and post it here.
Why did you use gasket sealer? If it is a standard bowl carb, remove the nut on the bottom and look at it. If you see little holes in the threads of it at top, clean them all out the best you can, and reinstall, and try it.
I know. I was being sarcastic.
The Briggs engines had points until up to 1983 or so, and I always recommend putting the newer style coil on, and just clipping the wire to the points, and leaving the flywheel alone.
I am not familiar with the saw, but if it runs on choke then dies when you move off choke, then it is likely a fuel/air issue. Electrical, it would have to ground out the coil when on run, but not on choke. Here is the wiring and throttle control blowup.
use a t-27 torx tipped wrench and remove the gearhead, just 1 fastener to loosen. See if the gearhead is bad/stripped, see if the driveshaft is still intact.
Here is a pic of the gearhead.
Take off the cover between your legs, where the forward/reverse lever is, so you can watch the belts/pulleys. It will not want to go in F or Rev if the belts are still moving, so check things out.
Yes, the units have had some minor changes, the throttle cable ends, and slight changes in the air filter assembly, handle, etc.., but the 4140 powerhead has been pretty consistant. The throttle cable end change has forced me to include the throttle cable with the carb on my listings.
Also...
Have you tried just richening up the mixture screws? The "primer" looks to be operating OK. It is actually a "purge" and pulls fresh gas through the carb and returns the excess to the tank.
You might just pull the limiter caps and richen up your screws.