Great summary.
QA testing can be extremely repetitive and monotone and not something most players checking out the alpha/beta will do.
Regular players on the other hand has a tendency to find smaller things that break other parts of the game, like lore believe it or not.
I was in the early MoP beta and most of my time was used giving feedback on how mob placement/names/behaviours affected my playing and giving general bug reports. Apart from that I explored the early game and found it fascinating to see how the game changed from a beta state to finished game. For example the Horde intro in Jade Forest was changed completely. You were originally supposed to go down with the airship and crash.
In short I joined the beta because I A) Wanted to have a look at the early game. B) Wanted to affect the game to make it better by coming with suggestions.
A QA tester often cannot give or aren't asked for suggestions, they are often just there to check if the game works. An alpha/beta with players however can give the developers different type of insight into how the game feels. Problem is that with hiring players to test you are not guaranteed to get players who are actually interested in playing and not just people who are in it for the money and don't give a rats ass about the actual gameplay and will just say whatever to get their cheque.
It's an exchange. Blizz gets testing and the players get to see and affect the early game.