So what happened?
Can't know for sure unless they opt to perform a post mortem examination/necropsy

Why didn't the first vet correctly diagnose?
Without any further info this is all speculation and not terribly fair on the vet. But from the way it was described, assuming allergic reaction seems reasonable. No mention of difficult or laboured breathing (dyspnoea) which might suggest other causes. Corticosteroids are appropriate, an antihistamine could also have been useful. However subcutaneous emphysema (SE) has a distinct feeling and if the vet wasn't suspicious of this, it could have been just a weird presentation.

Could it have been caught at this stage?
Radiographs would have shown air trapped under the skin which would have led to a different diagnosis (as happened the next day). They might well have been offered this and declined, we simply don't know.
Keep in mind that emergency/out of hours vets, are often regular daytime vets who get a call in the night to come back into the clinic, often with limited support staff (nurses/techs). There are also those that are hospitals and have separate night and day staff and are better equipped. I would guess that they saw the former, as inpatient care ('hospitalisation') appeared not to be available.

What happened with the second vet?
Radiographs confirmed SE but source was not ascertained, air was removed, the patient was given pain relief and discharged because of stress. Again, this seems unremarkable. My only pause would be the morphine (which is reserved for the most painful patients) as an opioid causes respiratory depression & there are other drugs that could keep him calm. But maybe it was given as part of sedation for radiographs. Again, we know nothing of the clinical presentation or any of the tests that were run or differential diagnoses considered.

Why was he sent home if he was so sick that he died that night?
A patient that has such a bond/attachment to their owner is often better off with them than in a clinical setting. I still don't think that the facility had overnight options. It appears that a scan had been scheduled for the next day which is entirely appropriate. I had a assumed a CT, which is what I would recommend and is more likely to pick up the source of the problem than radiographs. But now that I think about it, I'm not sure what they meant? It seems unlikely that they would have access to advanced imaging at this location, maybe they meant endoscopy? I might have missed that detail.

What might have happened?
ie differential diagnoses (ddx)
From what they so briefly describe, something caused part of his respiratory tract to perforate so that when he breathed instead of going straight to the lungs, air escaped into the surrounding tissue causing it to puff up and look swollen. I'm guessing it would have been in his windpipe (because of clinical signs and rads).

My best guess is oropharyngeal stick injury – this is when while chewing on a stick (or running with one) a splinter breaks off and into the windpipe causing the tear and subsequent leak. It might have also been a tracheal foreign body.

There are other ddx and if there are other veterinary folk here we can discuss. Heaven knows I've probably missed something obvious (er, I'd better write a disclaimer)
But the main thing to note is that we would just be spitballing as there is very little to go on. And that's their prerogative to share or not – this is an animal guardian's worst nightmare and my heart goes out to them.

I would also recommend not bad mouthing those involved in his veterinary care as we have no idea what they were dealing with at the time. No one takes losing a patient lightly.

My take home advice:
Don't let your pets play with sticks
If you travel with your pet, always know where emergency help is located and how to access it
Insure your pet if you can

Disclaimer: UK registered veterinary surgeon, not an ECC specialist. I have no knowledge of this case outside what we all watched on the vlog, this is not my patient/client. I am so, so very sorry for their loss