Thank you everyone for the suggestions! You've given me quite a bit to think about and quite a few leads to follow up on.
You've also prompted some new thoughts to float up in my mind. If you'd like to hear a newbie try to sort things out, read on
I agree completely with the practice/muscle memory/10,000 hour approach. 10,000 hours with the ukulele sounds wonderful, which is why I think I'm in the right place.
Music theory is a "big picture" that I'm learning, but not the "big picture" I had in mind. I can whistle without music theory and if I could play the ukulele as well as I can whistle, well that'd be awesome!
What I have in mind probably sounds much stupider. If I was a child with no knowledge of music theory, how would I interact with the ukulele? I'd probably strum it or try to imitate someone. If I'm lucky, I'd stumble upon some way of hitting the strings that sounds nice. And I would repeat that ALOT. Eventually I might notice that if I change it *this* way my noise sounds happier. If I change it *that* way the noise sounds sadder. One far off day, I might completely master making the sounds I imagine in my head on the ukulele. But it would all make sense relative to that basic strum I became comfortable with. That strum is the big picture I'm thinking of. It makes me wonder how self taught musicians learned. I've heard stories of blues men who started by playing the diddley bow as a kid (one board, one string, two nails). Maybe there's something there worth trying.
I probably need to work on my right hand. I can make notes happen with my left hand, but they don't really run together. I haven't come across much advice about the right hand, but I haven't really looked for it either.
Perhaps just one song, a simple song really well learned could be the big picture I'm looking for.
Anyhow, thank you again for the advice. This was a much bigger (and more helpful) response than I hoped for!