That brings up an interesting issue, which is that many systems do have more noise in y than in x. For instance, time series data from an analog-to-digital converter, where time is based on a crystal oscillator.
Well yeah, x is specifically the thing you control, y is the thing you don't. For all but the most trivial systems, y will be influenced by something besides x which will be a source of noise no matter how accurately you measure. Noise in x is purely due to setup error. If your x noise was greater than your y noise, you generally wouldn't bother taking the measurement in the first place.
You could, and maybe sometimes you would, but generally you won't. If at all possible, it makes a lot more sense to improve your setup to reduce the x noise, either with a better setup or changing your x to be something you can better control.