the 031 had a collection of small problems. it could be:
poor compression (worn rings, siezed rings, broken rings, etc).
worn points or lobe on the crank.
pinched fuel line by the air filter, rotten fuel line,plugged filter.
cracked impulse tube, debris in tube.
leaking intake manifold.
plugged spark arresetor.
worn internal carb parts, plugged screen in carb.
short in the ignition switch, wireing.
abuse such as oblonged end bearing from having the chain to tight as the saw cools and pulls on the end of the crank, but, that is abuse and a whole other world(wrong mix,air/fuel mix,etc,etc).
like anything else, without seeing the saw a proper diagnosis is not easy or likely. yes, start with the basics. compression,ignition and fuel delivery. when you find what is lacking find out why. marty