I was also a firm believer in Dude's maxim when I had one child, and it works for a lot of kids. But not all. I have also found that in fact giving a kid loving attention in spite of bad behavior is sometimes exactly what they need and that behavior then improves. People are more complicated than behaviorism acknowledges, I think.
I don't think it's a binary situation, in that it always works for some kids, and never works for others. Sometimes a kid's bad behavior needs to be ignored or otherwise unrewarded, and sometimes the same kid's bad behavior requires comforting and soothing. What is motivating the behavior? Is it a deliberate tactic, an attempt at manipulation? Is the child acting out? Or is this a cry for help?
In the example I described, the child had tried a number of tactics to stall or avoid homework, they hadn't produced the results she desired, and so she escalated. There was a degree of calculation involved. That sounds somewhat similar to the OP's description of fake injuries.
This method has served us well with our DD8 in a number of ways and situations, but whenever she's in an emotional state where she's completely out of control, we've abandoned it.
