kraftsman lawn mower runs for 10 minutes, sputters and dies

More than likely the problem will be solved with a new carburetor. Ethanol gas can plug ports and/or cause float issues. What you are describing is your fuel bowl filling up when sitting still, then you run it but the feed provides enough fuel to not keep it completely filled (or filled to where it's supposed to be). So after a few minutes of running, it dies due to fuel starvation. You reprime and it starts back up, runs for a little bit and dies again. If you let it sit for a long period of time, the fuel bowl will probably refill and you'd be able to run it again for another 10 minutes or so. I don't know if this is exactly what is going on, but will bet it is fuel related and then the carb is the root of the issue. I've found on small engines, it's easier and cheaper (when you consider the amount of time you spend working on it ... what is your time worth?) to replace the carb rather than trying to rebuild or cleaning it.

Just a guess, but it sounds like heat-induced vapor-lock to me. You didn't remove any sheet-metal shielding from between the carb and the motor did you? If so, put it back. And clean any airflow ports anywhere near the carburetor, to ensure that the airflow from the flywheel blows well over and around the carb. I'm guessing you're either missing some heat shielding or you have some gunk blocking the airflow and the carb is overheating.

If it's not an unvented fuel cap problem or a carburetor jet problem then it is most likely an electrical problem (coil, points, etc.) that becomes unstable when it warms up to operating temperature.

10 mins is maybe kind of long for this, but I'd still check for a spark suppressor screen in the exhaust. They can get plugged w/ carbon, and cause similar problems. You can burn it clean with a propane torch. (gently.)

The mower had been acting up for a couple of years. It would run for a few minutes, then after five minutes or so would start again. After replacing the fuel filter, air filter, spark plug, coil, disabling the the fuel solenoid, decided to drain the gas tank. I unhooked the hose at the fuel filter. Gas dribbled out slowly.

It turned out that the plastic spigot on the bottom of the gas tank had been twisted when someone had attempted to remove the fuel line. Ater installing a used tank from e-bay, the mower works fine. So yes, it was a fuel problem, but no it was not the carburetor.