I am not familiar with the design of your saw, but I had an Atom many years ago that had the same problem. The nitwit I bought it from had been lending it to an even greater nitwit, and the latter nitwit had been filling the oil tank with old sump-oil from his car, complete with a large volume of sludge. This had not only clogged everything up, it had also scored the sealing surface of the piston pump (that is, the outside of the piston, and the bore). It needed a new piston, and possibly a new bore (which from memory was part of the crankcase casting). I didn't attempt to fix it, the saw was garbage anyway, due to bad design and nitwit-damage in various other areas.
Of course if your pump is not a piston-type, this will be no help at all. The possible relevance to your case though is that you may have a worn-out piston, if it is a piston pump. The pump has to be able to create suction to prime itself, and that requires a decent seal between the piston and bore. There also have to be intake and output valves for that type of pump. A simple output restrictor may suffice for the discharge valve, but the intake needs a one-way device such as a reed valve. If that is stuck either open or closed, you will get the symptoms you are getting. On the other hand if it is a different type of pump it may not need a valve at all.
What we need now is a serious chainsaw guy who is experienced in overhauling oil pumps, to tell us about the mechanism they usually use. Any volunteers?
