Handing over the boarding card is the final step to prevent you ending up at the wrong destination.
I am the senior cabin crew member onboard the aircraft that I work on, and on average about twice a month the ground staff send a passenger down to my aircraft that belongs on another flight. Those pax are actually greatful that the mistake has been spotted BEFORE they ended up in the arse end of Europe as opposed to somewhere like Glasgow.
I remember one Friday night where I was boarding a flight to either Belfast or Edinburgh. There were 2 flights for my airline on neighbouring gates, and both gate shared a departure lounge. It was raining and the pax were having to walk accross the apron a few yards to climb the steps to my aircraft.
Now the vast majority of pax that day had done the correct thing and produced their boarding cards, showed them to me and walked down the cabin. I was suddenly faced with a surely looking bunch of around 10 people who insisted that this was a ridiculous rule and that they will NOT be showing their cards again and saying that I am making them get wet (pouring down with rain outside). I said to them that 1) it is not I that is making them get wet it is themselves for refusing to show the boarding cards. 2) I also pointed out that it is a DFT directive that all pax must when boarding UK airlines show their boarding cards as it is a final check to ensure they are getting on the correct flight. The pax responded with "Do you think we are stupid enough to board the incorrect flight? This aircraft is going to XXX" my response was "Oh really, well that's funny but I am going to YYY", they thought I was mucking around and I said "No seriously, we are going to YYY, you are on the wrong aircraft. Show me your boarding cards now or expect to be removed by airport Police!", one of them then pulled his boarding card out and it said either Belfast or Edinburgh (can't remember which destination I was flying to) but I then showed the group of pax my paperwork for the flight which clearly showed that we were going to an alternative destination. I made them stand outside the aircraft until the ground crew returned, after pointing out that they obviously had not been listening to the flight calls properly inside the gate area, I also then pointed out that you can not rely on the ground staff to check every boarding card properly owing to the fact that they are also busy reconciling cards to try to ensure the correct number of people are onboard and that I am the only person who can guarantee that they don't get to the wrong destination. When the despatcher returned and I sent the pax away with him, the front few rows of the aircraft were laughing and saying to me "Don't you get fed up with passengers like that?" I replied with "No, because the ones that always insist they are right and not stupid will always do something stupid or be proven to be completely wrong wthin a minute or two, I just stand here and wait for it to happen".
Remember, when a cabin crew member asks to see your boarding card they are just wanting to make sure you get to the right destination as my story above shows mistakes DO happen. Those couple of seconds at the door can save you several hours (or perhaps days) stuck at the wrong destination miles from where you want to be. And, think of it this way, you are on the wrong aircraft, aircraft crashes. Your family and friends won't know you are missing, you will be a "John Doe" as there will be no flight paperwork for you relating to that crashed flight. You might even in the event of terrorism related crash be incorrectly identified as one of the terrorists as you were not supposed to be on that flight!