Pro dog trainer here (with thousands of dogs worth of experience). This isn't typically an 'aggression' issue, nor is it an alpha issue. Most people mistake betas for alphas, believing that aggression means an alpha, but this is not the case, quite the reverse. Alphas know their place in the world and never have to enforce that.
More often it's that the older dog is trying to tell the puppy, "Bugger off and leave my shit alone!" but lacking words to say that, growls and carries on instead. And then the older dog catches hell and doesn't understand what it did wrong, and may get even worse because now the brat gets to harrass the older dog with total impunity.
Many adult dogs don't tolerate puppies at all (a few will even kill the puppy first chance they get); this is not unusual and you can't train them out of it. But you can make them less likely to react, and more likely to develop tolerance:
The solution is to make the puppy back off. Make it understand that it doesn't get to share everything, and not everyone wants to play with it. Not all toys are the puppy's to take whenever it wants. Don't force the older dog to accept an invasion of its existing space and possessions. Would you like it if someone thrust a roommate on you, who then took your stuff and pawed your face, and you have to grin and bear it? Of course not. A lot of dogs don't either.
Furthermore, not all dogs are social with other dogs. Many would prefer all other dogs just die, or at least leave them the hell alone.
Same applies when a strange dog wants to 'play' in public and does so by jumping on or getting into the other dog's face .... who is then very likely to growl and snap: "Hey punk, knock it off!" Would you like it if some stranger shoved you on the street? But which dog gets corrected?? ......Yeah.
Incidentally, an alpha dog may graciously share its toys, bed, etc. but when it wants them back, doesn't have to do anything more than stare off to one side, and the underling goes "Yes sir!" and departs without argument. Betas are the ones who fight among themselves, try to force and bully (like a human who is always trying to prove something), pick on the nobodies, and are much more likely to buck the human as well. But a beta will never sass an alpha. Alphas do not fight with other alphas, either. Alphas are generally tolerant of puppies, while betas are often not. -- And these social ranks are inherited, not made. I've mostly bred the beta traits out of my bloodline, because they are too much trouble.