The A6000 kit 16-50 lens? I really don't think it's that bad, especially considering the size.
The 1 inch sensor on the RX100 is a 2.7x crop compared to Full Frame. The APS-C is a 1.5x crop, so that's a pretty big difference. When shooting in lower light situations, you're definitely going to want as big a sensor size as you're willing to carry so that you can get less noise.
I don't think this really affects light intake (native gain) from the lens, but in terms of depth of field, F3.5 ~ F5.6 on a Sony APS-C becomes an F5.25 ~ F8.4 full frame equivalent and F1.8 ~ F2.8 on 1 inch becomes an F4.86 ~ F7.56 full frame equivalent, so the lenses aren't massively different in that respect.
The RX100 is super cool, and will probably serve you well, but I feel like the 1 inch sensor size is really starting to push it when it comes to image quality for the price.