While classic adventure (not action adventure) games are my favorite, I never like the extreme linear storyline games that Uncharted series and Last of Us use. There is no option to choose a different path or do anything different or even end the game with the bad guys that got away.
From most action adventure games, my first question is why do you have to shoot at everything that "moves" ie. kill all the bad guys as the only solving the problem to continue the story? It's like the moment the bad guys appear on the screen, you have no hope of ever escape unless you can kill every single bad guys that appears. What happens to an option to sneak your way out or reason with the bad guys? I consider having to shoot at every single "target" to let you continue is the weakest form of storytelling because you just "erase" the problem that is blocking your path. Isn't it more fun in solving the mystery on why it prevents you to continue your path than just "erase" it?
Second, able to change the story path along the way that changes the outcome in the end is far more fun since there are always more solutions in solving a problem. That's the reason it's an interactive adventure game, not a fixed story like a movie. Each player should be able to find his own path in solving the mystery which can leave the end with multiple good or bad outcomes. It gives the player multiple replay value of finding out different solutions and outcomes on the next replay. Also, games like Heavy Rain, it's fun to see how if you killed off all the good guys, how the bad guy can get away in the end. But with most games, if the primary character dies, it's Game Over. What happens to the game continuing on with another characters until all the good guys are dead which leads to the bad guy gets away as the ending? That is the area that movies cannot do. Even games like Uncharted series or Last of Us don't have because they only have one outcome in the ending with the main character which made them more like a movie.
You can say, play RPG games that have decently written story and outcome to the ending. While I also enjoy RPG games, they are designed mostly for action. Even the RPG game has decent of story, it can never match the complication of storyline in traditional adventure games. When what about JRPG? While JRPG like Final Fantasy series, the story and action often feel disconnected. Playing JRPG feels like play some action scenes, watch a long cutscene, play some action scenes, etc. And JRPG storyline still remain very linear with no option to alter the story path along the way. You can even watch the entire storyline in JRPG if you just extract all the cutscenes into one long movie. (Example, I found Final Fantasy XIII was sold as 4 DVD movie discs which contain only the cutscenes. It was sold as a movie, not a game. If I have never played the game itself, everything about the story is on the cutscenes.)
My only request of traditional adventure games is have longer gameplay. While I love Heavy Rain, the gameplay is very short unless I consider playing every possible path and ending are the total game. A problem with today's adventure games like Heavy Rain is the game is designed as too easy to solve. There is no effort in the gamer's part to solve anything complicated to go on in the story. In other words, there is no effort in the game that needs the player to put some problem solving efforts in finding a solution to any mystery along the story. It's almost like "go to the next screen" to see the next answer to continue on.