"What characters were in this game exactly? Between the four protagonists, you could maybe muster up two dimensions between them."
The goal of a game like this one, in which a good deal of the story hangs on the perceived peril of some of the characters, is to try to get the player to empathize with those characters. In this case, the developers tried to do that by giving you a sense of the familiar- the daily routine of the main character will, one hopes, let the player get inside his head a bit and be there with him as he goes through the tragedy. There not really no way at all to FORCE the player to have this experience, though. All I can say for sure is it worked for me- I got invested in the characters, and the story was intensely gripping because of that. If the game can't grab you, all I can do is assume the experience is going to be much less visceral and engaging. I do wish everyone could enjoy the game as much as I did, but in the end its completely a matter of subjectivity.
"Personally, I'd rather have bonded with the character because they had some interesting quirk or were funny, or had some good cameraderie with the people around them, or were mysterious in some way, but I suppose that will have to do. I have seen the light!"
Sarcasm aside, I'd personally argue that lots of the characters do demonstrate many of these traits, depending on your choices as a player. Shelby is pretty friendly when he asks questions, Jayden has a drug addiction somehow related to this weird high tech equipment he uses, your left to wonder about the state of Ethan's sanity, and so on. As above, its entirely subjective as to how much or little depth the characters offer the player.
"it's hardly an either/or scenario between "emotionally immersive" vs. "heads blown off for no reason", as there are numerous games that can get the former without being pretentious swipes from bad films (see: Silent Hill 2, Psychonauts, Ico, Shenmue, Assassin's Creed, hell you can argue GTAIV, the Metal Gear series, or the Final Fantasy series all do this)."
Two things there- first, there is quite a lot of action and violence in Heavy Rain. The fight sequences are pretty intense in my experience, for instance. Secondly, most of the games you mention draw their material from other sources- Silent Hill 2 uses stories from Steven King and concepts like Dante's Inferno (notice how the game is constantly 'descending' as you progress), venturing into mindscapes was hardly an original concept when Psychonauts came out, Ico uses one of the oldest plot devices ever told (save girl from castle) to drive its narrative, Shenmue follows the same plot of a dozen kung fu movies (those set in the modern day, at any rate), and Assassin's Creed is essentially Prince of Persia in a sandbox world with a clumsy Davinci Code storyline that mixes in about a dozen conspiracy theories. Most games are trying to emulate a genre told in other mediums. What matters is if it was executed well and if it offers surprises despite taking elements from other media. In my opinion, Heavy Rain does those things and does them well.