If FAs didn't check the boarding cards, why have them welcome you at all? People could just get on any door like a train, sit down, and then during the flight an FA comes to punch a hole in your ticket.
Personally, I like checking boarding cards as it gives us an opportunity to interact with the pax, understand their nationalities, who they're traveling with, whether they're drunk or angry, and gives me a sense of who's on board to tailor the service accordingly. Stopping/slowing the running stampede from the bus/gate also helps calm the crowd as they enter a plane (not a soccer stadium).
Yes, and it saves headaches if you can direct the pax who's bound for Brussels instead of Berlin to the right plane before we've pushed back.