Your results may vary, but I'm in the camp who believes, on average, that the snakehead period at Gibson produced mandolins with significantly different sound - tone if you will. Certainly departing from the teens era instruments. And also from most of the ones that followed. Eventually, it seems like the company completely lost the ability to even approach it's glory days, but let's leave those out of the discussion.
I've never heard anyone seriously suggest that the difference in tone was DUE TO the headstock design. It's just the most obvious visual sign of that era in Gibson's history. As for BETTER - I don't use that adjective around here. Matt Umanov once told me that no matter what an instrument sounded like, eventually someone will come in to the store and it will ring their bell.
For me, the snakeheads I've played have been a bit punchier on average, favored the high mids that help a mandolin 'cut' in an ensemble, and that's been true even of the several A-Jr models which have no truss rod. The other features of the era have been discussed in this thread, and are well known, and some made it into the 'post-Loar' era. I happen to like
Make of it what you will, but I don't think mine is an uncommon opinion among those who like vintage Gibsons. And I don't know that I can comment on what's the 'best deal'. I used to say the Gibson A-Jr snakeheads were a good bet for the sound that I like, but they now go for $2K-ish so I can't call them a 'bargain'.
Anyway, I'll close with two thoughts. The variety of tone and physical traits - and the fact that they made SO MANY from the teens through the '20s, is what makes shopping for a vintage Gibson A model so fun, you just have to find the one you like the best. And the other point - you may have to put on a new fingerboard if you want to play in tune. That's just the way it goes with SOME in that era.
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