In theory, it could if the camera used the right strategy for reducing the image size.

As you noted, with current crop-sensor cameras, the raw image remains the same no matter what JPEG size you have set. The JPEG image is simply scaled. This can somewhat reduce the appearance of noise, but the reduction is due to the image scaling algorithm (you can't fit as many speckly pixels into the smaller picture as you can into the full-sized version). It's more likely, though, that you'd be able to do at least as good, if not better, if you do the noise reduction and scaling yourself after the fact.

There is a strategy that will produce true noise reduction. Some high-resolution medium format backs (Like the Phase One SensorPlus series) use a strategy called pixel binning, where groups of adjacent sensels are treated as one much larger sensel and their cumulative charge is read from the sensor. That's different from reading individual charges and averaging (which is what you're restricted to in post-read processing) -- it occurs at the hardware level, and changes what "raw" means. The read noise has a better chance of cancelling out, and the cumulative charge makes the analog-to-digital conversion less ambiguous (the range of quanta converted is wider with less amplification).

In practice, this usually means cutting the resolution by a factor of four (half the width and half the height). With a 60 or 80MP medium-format back, that still leaves you with a 15 or 20MP image; with a 16MP crop-sensor camera, you'd be down to a 4MP raw image. Now you may know and I may know that a clean 4MP image is better than a noisy 16MP image, but not everybody will buy into the idea that it costs extra to produce a smaller image. That means that it's unlikely that you'll see pixel binning used in anything less than pro-level camera any time soon. It may appear in full-frame cameras if their resolution keeps climbing, but I wouldn't look for it in a crop sensor. (Well, maybe Pentax might take a stab some day, since they don't do full-frame.)