It looks like National Geographic is the culprit here for originating the .44 claim, for publishing an article in 2017 entitled:
Ancient Slingshot Was as Deadly as a .44 Magnum

I may be new to slinging, but I've been reloading for a couple decades.  Personally, I'd have to call BS on the .44 mag claim.  Sure, you might be able to cherry pick some random reloading data that "technically" backs up the claim, but that doesn't mean much to me.

A couple examples:

My only 44 mag is a Ruger super blackhawk hunter.  I shoot a pretty mild load in it because I'm not a fan of recoil.  Just checked those reloads and they are 245gr bullet chronographed at 1370 fps.  That converts to 6.61 Kgm/s.  Again, that is a mild load from a 7.5" revolver.  That same load in a 18" lever-action will probably clock around 1800-2000 fps.  So, maybe, you can make a comparison, but one needs to cite some specifics.

Another "in real life" experience occurred on my first black bear hunt in Ontario.  I took a 45 Colt lever-action with my handloads.  In a lever-action, the 45 Colt is as good as or better than the 44 mag depending on the load.  I don't remember the exact specs, but I think is was a 300gr bullet with a mv of 1800-2000 fps.  On about the 4th day of the hunt I finally got a shot at about 30 yards.  I hit and blew through that bear with that load.  I know that because, one, we had a blood trail, and, two, when I shot the bear he was walking in front of the bait bucket, which I also blew through.  Initially, I didn't know this.  Later, after investigating why the bucket was leaking, I saw two clear holes in the bucket.

I don't know how hard I can/could sling a stone, or other projectile, but I highly doubt I could sling it through a 250lb black bear. 

If I were ever in the unfortunate circumstance that I had to choose between having a .44 mag from a firearm, or whatever projectile from a sling, fired at me at 30 yards, I know which I'd choose.

Specifics matter in this kind of discussion.  The same load fired from a 4" revolver and a 20" rifle vary wildly.  Even if you limit it to revolvers, the results can be vastly different depending on barrel length and model.  A double-action S&W isn't as strong as a single-action Ruger.  That is why many of the "bear" or "+P" loads are "Ruger only" loads.  Again, specifics matter and I could be wrong.  I may or may not have consumed some bourbon before typing this response and that possibly could have affected my judgement.

Interesting discussion regardless.  Sorry for rambling.