BalloonSlayer you are mixing up genealogy with the principles of hereditary monarchy. As far as who could inherit the throne goes that only includes legitimate heirs. Except in the case of Henry the VIII it was absolutely impossible according to the law of the time for both of his daughters and he had them both declared illegitimate. In Henry's case what matters as far as inheritance of the monarchy was that they were named in the 3rd succession act, not legitimacy.
Henry only acknowledged one bastard, Henry Fitzroy, but probably had several more. To say he acknowledged his bastards is incorrect. He only acknowledged one and only then because he feared he would never have a legitimate son and was considering passing the succession to him. He didn't do it routinely.
Genealogy is more to do with genetic descent and legality. We can only prove (well, almost always) that people are descended from their maternal line. Paternity is much more difficult. As we can't routinely DNA test everybody assumptions are made based on who the legal father is (who the mother was married to or who recognised them as a bastard). But there is often doubt over that. There have however been recent legal cases over aristocratic titles which have confirmed that DNA descent takes precendent over believed or legal parentage.
There's been a lot of debate about the Carey children's parentage. The general gist of current consensus is that her son was probably not Henry's but her daughter Catherine was. This is based on knowledge of the dates their affair occurred and the customary expectation at the time that a King's mistress would not sleep with any other men including her husband.
So, yes, it is fair to say from a genealogical point of view that Kate Middleton is, on the balance of probabilities, traceable back to Henry VIII.