Corrections on your assumptions: the A6000 and NX300 have a larger APS-C sensor. The crop factor for 1" (aka CX format) is roughly 2.7x, while APS-C is 1.5x. 4/3" (or four-thirds) is 2x. And the lens on the RX-100 is actually 8.8-25.7mm f/1.8-2.8 lens, but the 2.7x "crop factor" gives it the "film equivalence" of a 24-70 lens--which only means the angle of view would match that of a 24-70 on a full-frame camera. The 16-50mm lens on an APS-C NEX or NX sensor has 24-75 film equivalence.
APS-C compact fixed-lens cameras do exist, though (e.g., Nikon Coolpix A; Fuji X100S, Ricoh GR, etc.), and there's even a full-frame compact (Sony RX-1).
As for how much better the image quality would be, that's a hard thing to answer, depending on how one judges image quality. It can be worth it, depending on what and how you're shooting. We expect larger sensors to be better at rendering a thin depth of field, a larger dynamic range, and better high-ISO noise performance, with the possibility of higher resolution. If you're shooting landscapes at low iso and deep DoF, without an extended dynamic range, maybe you won't be able to tell any difference between formats. How much the difference between the formats matters depends on what you're looking for, and how much that difference is worth to you in dollars; and that's a personal judgement call. I think most of us here would say yes, it's worth it. But then most of us have already slapped down thousands of dollars on system camera gear and use it on a regular basis.
But there are reasons dSLR shooters buy an RX100III: for size, portability, and convenience. A system camera, even a mirrorless one, still requires that you cart pieces of that system around. Probably multiple lenses. This gives you added versatility, but adds to inconvenience.