Why is the Captain's intervention supposed to work any better than, say, the purser? I mean the man has training to fly a plane; not to be a bouncer in a bar!
You're right in that the pilot is there to fly the plane, however they are in charge of the aircraft,m everything on it and everyone sitting behind them.
In days gone by when a captain spoke to a passenger they would have been seen as a figure of authority and the passenger would almost certainly have shut up and sat down and stayed that way until the plane landed. Nowadays the pilot is just seen as another target for some of the idiots who fly. Someone else to argue with, lash out at or ignore. TV shows like Airline have shown people that if the break the rules, they can still get away with it, or if they shout and scream at staff, they'll still be allowed to go on their way. It's always the airlines fault, always the crews fault, no-one is able to take irresponsibility for their own actions any more.
Thankfully the vast majority of passengers are perfectly well behaved individuals who do follow simple instructions.