I'm adding a 220 volt circuit. It appears to me that the breaker (20 amp, 2 pole in a GE panel) is only going to pull power from one of the two bars in the box. For 240, wouldn't the breaker need to tap into the bars on each side? Am I over-thinking this?

Thanks.

I believe you are overthinking. If it is the right breaker for the panel, it will produce 220 volts. If you were to put two single pole breakers side by side, they will tap into two different phases and produce 220 if you were to measure with a meter.

Thanks.

Your are correct that the breaker must tap into both sides of the panel to give you 240 volts. A double breaker does that. The panel is what's called a split buss panel. Any two spaces down one side of the panel come from opposite legs of the service.