I agree with Mike. The bystart system is defaulted to "cold start enrich" when not powered. While anything is possible, like Mike said, if there were a problem with the bystart valve, it would typically be stuck on rich and the engine would still start easily but then run poorly as it warmed up. Also, in over six years of dealing with these bystarters, I have yet to see one go bad.
But, then, it could be stuck somewhere "in between" so trying a new one, like you said, is cheap and COULD be easy. I say COULD because you sometimes have to remove the carburetor to take it off and put it back on. Not sure about yours as I have not worked on that model.
I don't profess to be an expert on these since I only work on the twins with single carbs so take my suggestions with a large glass of skepticism. But, note that there is only one bystarter for three carburetors. There is some tubing, I believe, (again, haven't worked on this model) that connects the enrichment circuits for the lower carburetors to the upper one having the bystarter. You may want to inspect those carefully for cracks,holes or blockages that might affect the enrichment system before replacing the valve or digging too deep.
All of us here though know that Mike is usually right and you may be looking at a good carb cleaning to "clear things up".
I don't consider it a MAJOR fault but it is a fault nonetheless in that the Honda carburetors of this era are a bit finicky. It sure sounds like you have been doing all that you can to prevent problems but may still have fallen victim to the "clogged carb".
I did not see any mention that you DRAIN them regularly. This is VERY important with this series of carburetors because the passages in the bottom of the float chamber are so critical in the proper flow of fuel through them. Draining them FREQUENTLY is the only way to ensure that debris does not collect in the float chamber passages. These are GREAT outboards so I hope you don't have to go through too much pain to get it fixed. And, when you do, add carb draining to your already excellent fuel maintenance routine.