I definitely want to see the same protagonist across multiple games. As has been said in this thread, it allows you to make a much stronger connection to the characters if you get to deepen your relationship with them mroe and more over years. That's what made ME so special. Garrus wasn't just a great character we met in a game. He was a great character we met in a game, then met up with again in a second game and got even closer to (possibly even romancing him), and then finally teamed up one more time as the best bros of the galaxy. When you met him in ME3 it felt like reuniting with an old friend, someone you had known for years. And that's an awesome experience that hasn't been replicated in any other game.
Sure, there are pitfalls when it comes to making a trilogy with the same protagonist, as you have to account for more details, and are perhaps expected to keep a higher degree of consistency between games. But they can learn from what the missteps of the previous trilogy, and even with a few missteps of their own I still think it's worth it.
Series with new protagonists in each game also have problems. Take Dragon Age for instance. When Leliana, or Cullen, or Morrigan show up in Inquisition, I already know them as a player, but my character doesn't, and has to be introduced to them again, and ask them about all the stuff that I as a player already know. That creates a disconnect between me and my character, which harms roleplaying.
Not to mention that the problem of honouring player choice throughout games is not unique to series with the same protagonist. DA has a new protagonist in each game, and even largely takes place in new locations, but that doesn't mean that some people are less than happy with hoe poorly some choices are reflected. Made Dagna stay in Orzammar? Doesn't matter. Still turns up in Inquisition. And ME2's plot largely treading water is not because it Shepard was the protagonist throughout all three games. It was simply the writers not planning very well.
There is nothing that says that new protagonists each game make for better writing or better representation of choices, it's all a matter of how the writers deal with the respective problems of each way of telling a story.