A custom device including one knob to control the angle and one (physical) slider to control the strength. I know this is stupid when thinking about a casual game, but it would totally make sense to me if my job was to "play angry birds 8 hours a day".

My point is that proper "physical" controls are better for almost every task out there. The obvious downside is that you can't make them appear and disappear depending on the current application being executed ;)