The magic baby hold, as demonstrated by a doctor here, is a face down, arms crossed, 45 degree angle hold with a little bit of neck rubbing thrown in for good measure.
The big secret is is stops a baby from flailing arms, whacking themselves in the head, and otherwise doing the physical movements that lead to the breakdown that will be coming.
I used to use a modified version with the football hold, although without the crossed arms they can still flail about like a mad baby if they’re particularly upset. I’d use my other hand to hold the flailing arms for a minute until calm prevailed.
Alternately if you need to calm them and put them down, swaddling has the same effect. Aerin even smiles as she’s being swaddled.
Generally the hold as demonstrated is good for getting a melting down baby calmed quickly, but if you need your arms for anything it’s not much different from a wrap that keeps the arms down and close to the body.
I think anyone who swaddles to calm probably looked at this hold and has done something similar.
Next “magic baby hold” style technique I expect to hear about is how to calm a fussy baby who’s so angry they won’t feed by blowing gently on their forehead. “Oh wait, what’s up? I’m getting what I want… ohhh… nevermind then”. I’m sure it’ll be dubbed the kiss of life or something similar.
Not that there’s anything wrong with the doctor’s technique, but it can be done with one arm, or a baby on their back, plenty of room to branch out as there’s not too much limiting you other than baby safety.
Of course those both seem like direct ripoffs of various hippy parenting guides I’ve found from the 70’s. perhaps this is just a rediscovery, who knows? It’s not new though, but now you’ve got a video that can show you how to do it, and that’s a good thing.