Hi mills,
No experience with these but, after looking at the parts diagrams, I agree with you that there's no coolant passages in the cylinder head. Also, I don't believe that the water pump is intended to supply cylinder barrel cooling either as both head and barrel appear to be, as you've said, air cooled....judging from the cast fins in both.
I suspect that the water pump is probably solely intended for exhaust cooling only. I'm not sure how water from the pump would enter the crankcase through exhaust but, I suppose, anything is possible.
For water intrusion into the oil, on an outboard of this design and age, I might be suspicious of the oil and water seals that are "stacked" against one another at the bottom of the oil pan. See items 11 and 12 in the link below.
If both those seals were failed, water could "push" up through both as the outboard moved through the water. Especially if it were mounted very low on the stern.
Although, if that were the case, I would also suspect an oil leak at that location when the outboard was stationary. So, just guessing about the possibilities here.
The good news is that both seals are still available. Not true for MANY of the repair parts for this one any longer.
You might also want to investigate the water tube to engine connection in case my theory is all wet and water is indeed entering via the exhaust system. Items 4,5,6,and 7 in the link below. None of those seals appear to be available but could possibly be fabricated if there's enough left of the old to copy.
Another possibility that comes to mind is that water might find it's way to the oil via the crankcase breather system. Items 6,7,8 and 9 in link below.
The valve and gasket are still available too!
Out of "ideas" but I will keep thinking on it.
Meantime, here's where you can purchase the shop manual if interested.
Be sure to match up your vin number before ordering as there are more than one offerings.
publications.powerequipment.honda.com
Good luck.