Agreed, priming a problematic starting saw is my first step as well, it often gets the saw started even if its only a couple seconds. From that initial startup much can be determined but by priming one can get around a bad carb or air leaks , they can be determined after a saw starts. If the saw starts and quickly runs out of gas, stops then it is likely a fuel delivery issue, bad fuel line,filter, impulse line or a carb that needs rebuilding as the diaphragms are hardened. A partially plugged up screen in the carb will usually allow a saw to start and idle but not allow the engine to rev up, a cracked impulse line will do the same thing often, a cracked or broke off fuel line won`t pull fuel very well, this gets tricky some times if the line is broke/cracked in the tank itself, the carb can still pull fuel if the fuel level is above the crack but when the level changes and the crack is exposed to the air then air will be sucked into the line. If the engine will not start after a prime then it either does not have spark at the right time or its flooded, test for spark and at the same time it will be evident if the engine is flooded, while pulling the recoil to test for spark often fuel droplets will be spit out the sparkplug hole. If the engine has spark then drying out the engine and reinstall the plug should get it going again if proper starting procedure is used, if the engine floods again and did not start then the carb needs attention. Bad seals and air leaks won`t cause a engine to flood, they cause such a lean condition it can make starting them difficult, if the engine starts with an air leak it will run at a high idle as the fuel to air ratio is off, more air than fuel causes higher RPM with a loss of power on top end. Pressurizing the engine by any means and applying dish soap around leak prone areas will most always locate air leaks by blowing bubbles, only case I can think of not finding a slight air leak this way is if the seal is only leaking under Vac, pressure is causing the seal to push outwards and seal whereas under Vac the seal is sucked inward and loses contact to the crank causing a leak.
With all this said and done I firmly believe in doing a Vac and pressure test on every problematic saw as I have encountered so many that exhibited sporadic malfunctions that one would be chasing their tale and throwing part after part on the engine trying to guess what might be wrong instead of diagnosing the real problem right off.