If my memory serves, the method I'm thinking of to make a single cylinder engine self starting has something to do with putting pins in the flywheel with a set orientation to TDC & BDC and then having a linkage connected with the throttle / reverser valve such that when the valve is opened the linkage pushes on the flywheel pin enough to ensure that the engine is moved off TDC / BDC in such a way that it can have a chance at self-starting. I believe the set-up that I recall was servo actuated, so the linkage was perhaps a bit easier to hook-up.
I haven't found the actual bookmark to this yet, but given the number of steam related bookmarks that infest my computer, that is not at all surprising. ;c)