If the G1 X had a prime lens, say 35mm f/1.8 or 50mm f/1.8 (which would be beyond 80mm equivalence - it's not a simple crop factor anymore because the new sensor is a different image ratio, I believe), the lens would still be too large for a compact camera - in fact it would be larger, because not only would the front element still have to extend as far from the sensor (60.4mm), it would have to be much wider around, meaning that the lens would probably have to be wider around than the camera is tall (considering that even on many "fast" primes the front element is nowhere near the total diameter of the actual lens housing)!
The essential fact of cameras with smaller sensors, while they may have fast apertures and narrower fields of view (same nominal field of view as a longer focal length on a larger camera), there is still less depth of field control. And while you can get an equally "fast" aperture with a shorter focal length while keeping the same field of view with a smaller sensor, the total amount of light hitting the sensor is only equivalent to that of a longer focal length lens at a smaller aperture in a system with a larger sensor - meaning that if you have the same focal length, the larger sensor can receive more light even with a smaller relative aperture (remembering for a moment that f-numbers are a ratio, not a fixed size).
If the larger format has a longer focal length and the same aperture, it actually is at a significant advantage in terms of the amount of light reaching the sensor. This will mean that you get greater control over depth of field (see how deep depth of fiels is with a wide lens versus a super telephoto of the same focal length) and, crucially, the image can be cleaner.
Going to a small format does not give you a free lunch here.
Since it has a sensor closer in size to APS-C cameras, to get the same depth of field and light intensity you need a lens much closer in size to APS-C cameras. The genius of the G1 X is that it is taking what is essentially the usual DSLR and kit lens combination and putting it in a smaller and probably more rugged package.