I suspect that, for looking as far back as Richard III or Thomas Jefferson, nothing but the Y chromosome (or, for a female ancestor, the mitochondria) would be of any meaningful use. Everything else would probably be about an equally good match for anyone in the general geographic region. It’s not a matter of how quickly the chromosomes mutate: The mutation rate for any chromosome would be slow enough to not be a problem over mere centuries, and I’m not sure that the rate for the Y is even actually any different from any other one. The real issues are assortment and crossing over.
For every one of the 23 chromosomes, you have two, one each from your mother and your father. The one from your father might have been the one from his mother or the one from his father, chosen randomly, so if you compare, say, your chromosomes #3 with your grandfather’s, there might or might not be a match. For someone as close as a grandfather, it’s overwhelmingly likely that there are going to be at least some chromosomes that do match, but as you go further back, it becomes less and less likely. And in most cases, there will be no record of which one you’d expect to have gotten, so you can’t even tell if a lack of match means anything.
The one exception is the Y chromosome, because that determines something that’s very obvious and known to the record books. If you’re a man, you have a Y, and you know that it came from your father, not your mother, and you know that he got it from his father, not from his mother, and so on.
And actually, it’s even worse than that for most chromosomes, because you’re not necessarily dealing with a complete chromosome. The two chromosomes in a pair can cross over, swapping the corresponding genes, resulting in a new pair of chromosomes that doesn’t actually match any single chromosome in any single ancestor. Again, this happens with every chromosome pair except for an X-Y pair, because they mostly don’t have corresponding genes.