I think the phrase "alpha dog" is overused and overrated.
Great opening line to a speech or paper.
I agree. While there are alpha dogs out there, meaning dogs that, when living in a family with a group of dogs, are naturally the leader of the pack, usually this type of dog never has the label of "alpha" applied to it.
The dogs that are scrapping with each other are usually beta dogs, insecure and trying to assert their position through force. A true alpha uses their confidence and body language and rarely needs to growl or snarl at another dog.
And alpha dogs are not the ones causing all the problems with dog-human relationships. Yes, you can be over-matched by a dog. But most stable dogs have no problems existing even with less confident owners. Usually problems arise when less confident owners, are matched to dogs that are less confident, who would do much better with a sure, strong leader. Then as the dog lacks trust/confidence, his behavior becomes scary to the owner, whose confidence diminishes further, as his fear increases, and the dog becomes truly dangerous.
And, thus, all the focus on dominance and alpha in training and behavior generally leads to more confused people and dogs. And while some owners may become stronger leaders (have some structure and consistency) and inspire a semblance of confidence in their less than confident dogs, many others of these dogs will have even more trouble by being shown who is the boss, and being ground under toe with whatever the current dominant/alpha training/behavior modification calls for.