7D or 70D: these are fast enough, but do they do well enough in low light? I'll define "well enough" as having at least the same image quality at ISO 3200 as the Rebel does at 800.
6D: fixes the low-light issues, but is the frame rate fast enough to have a decent shot at capturing action?
I'd appreciate any advice on this. Should I go for one of the above options, or just bide my time and save up for the 5D III?
Fundamentally, ISO/noise performance is a factor of two things: Real sensitivity (quantum efficiency) and total
sensor area. Assuming you frame your subject the same, the only way to really reduce noise is to use a larger sensor. Pixel size does not really play a role unless your only putting the same number of pixels on the subject (which means you are not necessarily framing the same). With the same number of pixels on subject, then pixel size matters, and larger pixels do better.
Given these facts, it's highly unlikely that you will ever see an APS-C camera that has ISO 800-level performance at ISO 3200. Even if Q.E. reached 100%, your still not going to see that much of an improvement. I think the XS had around 25-30% Q.E. The 7D has 41% and the 70D has 45%. That is not even a factor of two improvement over the XS, let alone a factor of four improvement. You will see an improvement in high ISO IQ, but no where near enough that ISO 3200 looked like ISO 800. ISO 1600 will look a lot better, almost as good as ISO 800 on your XS, but still not quite as good.
The only real way you are going to get a significant improvement in high ISO performance is to move up to a full-frame camera. The 6D has about 50% Q.E., so just from a real sensitivity standpoint, it's about twice as sensitive as your XS. On top of that, it's got 2.6 times the sensor area. So you'll gain almost two stops of improvement in high ISO performance...indeed, ISO 3200 would look a lot like ISO 800 on your XS. Because of the greater pixel count, if you frame the same, ISO 3200 should look quite a bit better than ISO 800 on your XS for a given output size. These same fundamental facts apply to the 5D III as well. It has 49% Q.E. and the same total sensor area.
Whether you choose the 6D or 5D III is really up to you. The AF performance is indeed much superior on the 5D III, and for action, that will certainly be better. The 6D's 11pt AF system is not bad, though, and at least for kids and for some lower action wildlife, it should suffice quite nicely. The 5D III's AF system is going to be able to lock onto active subjects, like kids running around or wildlife in action, a lot better than the 6D's. For action, frame rate is pretty important as well, and I'm not sure the 4.5fps of the 6D is really quite up to snuff. Personally, my limit, having used 3.4fps, 3.7fps, 6fps and 8fps cameras, is about 6fps. Anything lower than that, and I really feel I'm missing the right moments. At 8fps the frame rate feels really good, and you definitely see an improvement in small changes in subject pose and orientation that give you the option of picking a really ideal frame. The 7D, which I own, has an AF jitter problem that causes some frames to become very slightly out of focus, which basically negates some frames for all but the smallest reproduction ratio online. My average "keeper rate" is about 5-6fps at best, which is part of the reason I feel the 6fps of the 5D III, with it's much better AF system, is acceptable for action. I say acceptable, however ideally I'd say 8-20fps is probably where the sweet spot is, and I think 10-12fps, maybe 15fps is really quite ideal (assuming the AF system can keep up.)
At the very least, given that you've stated high ISO performance is one of your biggest concerns, that you should really move up to a full frame camera. The 6D is a very nice camera, and for certain applications, like landscapes, low light photography, and still scenes (like floral macros), and even astrophotography, it is a very, very good camera. For action, it's maybe middle ground at best. The AF system is capable enough, but the 4.5fps frame rate may just not quite be up to snuff. Whether you choose the 5D III or not would really have to be up to you. If you do choose the 6D, I honestly think you will find it to be a truly MASSIVE improvement over your XS...despite the shortcomings of the 6D vs. the 5D III, it really is a vastly superior device compared to your XS, and once you get it in your hands, you'll understand what I mean.