1. burst rate/ buffer handling
2. focus / tracking ability
3. reach
4. low noise signature at high ISO
What's "reach", in your view? The term is so vague and depending on whose opinion you ask seems to relate to some combination of magnification, the physical overall size of the sensor and the size of the individual elements on the sensor. Magnification is entirely a function of the lens configuration; it's going to project the same image at the focal plane no matter what the size of the sensor of piece of film placed on the plane, so we can discount that right off. The physical size of the sensor just governs how much of the projected image at the focal plane we sample or crop, be it a 36x24mm rectangle for full frame 35mm or 24x16mm for a 1.5x crop (yes, that's Nikon but the math is easier).
That just leaves the element size, which is actually providing additional detail within the image and not any additional focal length implied by the term reach. True, the more detail you have the larger you can output the uncropped image before you notice any pixellation which equates to a lossless "zoom" at the output stage, but that's totally independent of the sensor size. With current technology you could hypothetically have a FF sensor with a 12um pitch and a 1.6x crop sensor with an 8um pitch - everything else being equal, which would you prefer?
How does going full frame positively affect any of these things?
In other words, what benefit would the sports shooter get when
going full frame? They lose added reach and incur more noise. So
why would they want it?
How does it negatively affect any of those things?
Burst rate is currently limited by two things; the mirror mechanism (if present) and the ability to spool data from the output buffer onto the memory card, and the latter can be mitigated by providing a bigger cache - DRAM is cheap after all. Even at 10fps, 20MP 16bit resolution you only need a 400MB/s bus from the sensor to the buffer (a few technical issues aside), which is not exactly a huge amount by current data bus standards, but still 10x more than even Sandisk's brand new Extreme IV cards can handle.
Focus has absolutely nothing to do with the image sensor, how can it since it's hidden by the shutter until exposure? There are actually some dedicated sensors in the base of Canon's DSLRs that perform this function in conjunction with the lens and the onboard DIGIC processor, but that's another subject altogether.
Skipping reach for now, noise is mainly a function of the size of the individual sensor elements, not the overall size of the sensor itself. Think of each element as a bucket collecting raindrops (the photons) - the smaller your buckets the more prone they are to overflow, possibly into adjacent ones, and the less likely they are individually able to provide an accurate indication of the rainfall. Want to know how much rain fell in an hour, but don't want to wait, then instead of waiting for an hour, why not wait half an hour and then double the amount of rainfall you measure in the bucket, or even quarter of an hour and quadruple it? This is essentially how a DSLR simulates the higher ISOs of film which does genuinely react to light faster, however in doing so you also multiply any errors, which is why higher ISOs are noisier.
The upshot of this is that the smaller you make the elements, the lower the number of photons that are going to strike them in a given period of time, and the greater the margin for error, especially when you start increasing the ISO. Ironically this means that the very thing needed for "reach" as I think you mean it, namely smaller sensor elements to provide additional detail in a subject at a given distance from the camera, will mean paying a price in noise levels, just the opposite of "they lose added reach and incur more noise".
As a sports shooter myself, I would much rather see a 1.6 crop body
with faster focusing, better burst. I don't see what benefit a
full frame would give me.
To be fair, possibly none. That would depend on your shooting style, how close you are able to get to your subject matter and your willingness to experiment with the creative opportunities afforded by a larger sensor that doesn't crop your image for you. You might have seen shots taken from right behind the net at football matches - many of those were taken remotely with a ultra wideangle lenses on 1DmkII and even 1DSmkII bodies, for instance.
Here's something for you to consider though. It's pretty much a given that the body most commonly seen in the hands of professional sports photographers at any event will be the 1DmkII (or mkIIN). If "reach" is so important to sports photographers, then why are they all using, and in the main perfectly satisfied with, the camera with the second lowest pixel density of all Canon's current DSLRs?
[For the record, from the lowest to the highest, the order is 5D, 1DmkII, 350D, 1DSmkII, 30D.]
Andy