The answer is: fast and accurate Autofocus, that is sensitive and do not hunt in bad light.
For a P&S, this will be the thing making or breaking the pictures.
Back to topic, a bigger sensor (by 2x), equals a stop faster lens. All things equal (noise processing etc.) it should always be able to have a stop higher ISO with same noise, thus enabling same short shutter speeds as the brighter lens. But it will always have more flexibility at base ISO.
In reality, this scenario is very rare, so go with whatever feels right. There are more things to a camera. Color output, flash, handling/ergonomics and so on...
But my (admittedly biased) view of wives in general, they value responsiveness, simplicity and end IQ. So, G1X with large sensor, but complicated controls and slow AF would maybe not be a good buy, even if it has a large sensor and good IQ.