To be honest, I don't think this is really exclusively a video game issue. Movies, TV shows, books, comics, and other forms of entertainment all utilize the Damsel in Distress (which I'll just shorten to DiD from here) trope, and have been long before Video Games have existed. It even says that in the video in question. So the point of this isn't "Omg Video games are evil sexist things". Is the trope overused? Definitely. However, I don't think it's quite as damning as they'd have us believe. Media of any sort can influence people, but it's only just that; influence. I believe that's what Jb was implying; just as not all people who play GTA will turn into violent killers, not all girls who play video games will be shoehorned into the traditionally "weak feminine" archetype. Media is just one influence, and it comes down to the individual to accept what they see, or reject it.
Some people are more susceptible to the media influences, and they're the ones most susceptible to the effects of stereotypes. Just as many, however, will play the games, and just reject what they see. Not every girl who plays a Mario game is going to suddenly turn around and think "I should be like Princess Peach because that's what this game is telling me".
Even so, I think the take home message here isn't "You shouldn't play these video games because they're sexist and that's bad". It's more a matter of wanting more diversity outside of the typical DiD/love interest/fanservice object tropes. And while I'd agree that a little more variety would be nice (especially with Mario, I found the plot or lack thereof for the New Super Mario Bros. games to be terribly lazy and uninspired), again, I'd have to say it's not quite the end-all damning effect on females that some might make it out to be. But that's just this guy's opinion, and I'm prepared for any backlash that might come my way as a result of saying this.
That said, I don't necessarily agree with all of the examples given, especially the Shiek -> Zelda one. But, that's another story altogether.