This tree was on an island all its own.
I put the new batteries in the boat, and it started fine with that improvement. One problem solved.
We took the boat out for a short spin (not a fishing venture, just a short trip) and found another lurking issue, however. It’s the same problem we’ve always had (since I’ve been up here), though it’s very sporadic and inconsistent enough that it’s hard to reproduce on demand: sometimes when the main outboard engine has been running a while, it begins “hiccupping” – it’s a symptom that resembles the vapor-lock I’d get on my old VW Bug back when I had one. But it’s not vapor-lock, because the engine is not actually running hot. Chet (the boat mechanic in town) has suggested a fuel line or fuel quality problem, which is plausible but unprovable and hard to test or fix.