An intermittent spark could indicate the magneto is on its way out. I've never worked on a magneto, so I can't help you much there, but if you google "magneto rebuild", there should be plenty of assistance that way. Another thing to check is the HT lead between the magneto and the plug, is that good? If it's original, there's every chance it's wrecked internally (if not externally) and could be contributing.
I'll assume that your carburettor is OK, based on the fact you've cleaned it up, although depending on how deep you've gone into it, you might need to give it a full rebuild.
Other things to consider - what's the compression like? If the rings are stuck, or the valve seats aren't good, you might be getting marginal compression, which won't help starting a cold engine. Valve seats you can inspect by removing the head & simply pushing open each valve. Any pitting on the seats will require grinding out.
In the brief run you did get out of it, how did it seem? Was it a bit huffy, or weak seeming (both would point to compression)? Was the exhaust clean, or was there evidence of grey/black smoke (over fuelling)? Did it get murder hot (running lean)? Was it responsive to throttle changes (iffy carb if not)? Was it misfiring sometimes (magneto, lead or plug)? Something else to consider, if your spark plug is not of the correct heat range, you might be having problems with it. Did you pull it out after the one successful run & get a look at it? Assuming it's not since been washed clean by unburnt fuel, it may still tell you something about how the engine ran that last time - e.g. did it get fouled, or burnt up, etc.
FWIW, my experience of petrol stationary engines is limited to one extremely temperamental Villiers engine, which I loathe, but which one of these days I really ought to fix...