FYI: I have no further information about this "machine gun", but find it interesting.

Way ahead of it's time. Another Gatling type gun, but not a MG.

Tench Coxe concurs

Their swords, and every other terrible implement of the soldier, are the birthright of an Americans.

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If I recall correctly, this gun or one very similar is what Sam Colt modified and shrunk to create the handheld revolving pistol.

And yes I would have loved to have seen it used against the redcoats. There are many other multi-shot weapons of war that were created but most were not practical or were cost prohibitive, often both, not to mention so heavy that it would be a crew served weapon that was not as powerful as a cannon.

Ian talked about this gun in some detail back at the end of 2016, a nice little Christmas present.

The idea of using a revolving cylinder to have more loaded shots available dates way back.
The idea of mechanically indexing that rotation took some time, and some large improvements in machining to materialize.
The idea of having the axis of rotation paralleling the bore was not universal, either. There were a number of "turret" designs out there, both horizontal and vertical. (Mind, having loaded cylinders facing the firer did give some pause, as chain fires were a thing in those days before machined powder became common.)

Sam Colt had the benefit of a number of innovations coming together at just the right time.

I’ve been trying to find a diagram of the extended mag air rifle taken on the Lewis & Clark expedition, and described in the Mag ban briefing by David Kopel and others. There were some ingenious designs way back then if you had the $$.

Don't know if you've seen this page, but Beeman's has put together quite a thorough description and set of photos and drawings of the Girondoni air rifle:

https://www.beemans.net/Austrian airguns.htm

Didn't seem to have a lot to do with having the $$. These were actually military issue air rifles, seeing at least some actual service in the field. They were eventually deemed to be too troublesome to maintain for the common soldier and abandoned, but not before some 20 or so years of use in the military of the day. I'd imagine the limitations of leather as sealing material would be one of the worst problems, requiring just the right lubrication and occasional replacement of seals. Simpler to issue powder burners.

And this fellow seems to have built a pistol based in part on that system:

https://ctmuzzleloaders.com/ctml_experiments/lorgir_pistol/lorgir_pistol.html

Nice point. Maybe the Puckle might cause some people we all know to pucker.