It is not just a game, it's an experience!
In short, I did. Though not at first. It took me about 3-4 hours into the game before I start to care about the characters, making sure they're alive and all. I felt a twinge of sadness when Jayden died in the car junkyard because I was not quick enough to fight.
Which brings me to the control format. Yes it's a bit like a QTE but it's so innovative that it mirrors what's happening on screen. This is the way all new adventure games should strive for. A combination of good plot, and innovative interaction.
Heavy Rain is truly an experience, and despite no sequel coming out (according to the developer), I'm looking forward to more such adventures with other different plots based on this platform.