I've seen several Stihls that the fuel line would suck shut. It seems worse since we have ethanol in our fuel.
One other thing: the problem could also be a leaking crankshaft seal. The leak may not occurr until the unit is on its side and the crank moves over a tiny bit. (And the leak itself may be due to worn crank bearings.)
I would do a compression test on the saw before I did anything else. Pull the rope until the gauge quits rising. If it goes to 130-150 psi. it is worth trying to repair. If less, there is a problem with the piston/ring/cylinder.
Some Stihl saws may run at 120 psi, but may be hard to start when hot and the compression is even lower.