Warded Ms. MacDonald/Mr. Malfoy.
I'm well aware of the fact and I have been in school with many of your relatives. While lineage is not dispositive, the Hat is supposed to sort children based on personality traits and not personal choice, traits that are usually cultivated by the families. Even when personal choice is an issue, most tend to want to be like their families.
I don't think that Dumbledore can overwrite what the Hat does. I do think that he hates anyone who is sorted into Slytherin on principle. That will not change.
You give me too much credit. The choice was made by Salazar Slytherin himself that no Muggleborn should be sorted into his house. In fact, he wasn't too kind to Half-bloods, but he accepted those who were ambitious enough. The Hat follows the wishes of the Founders. No board or Headmaster can change that. There will never be a Muggleborn in Slytherin for as long as the Hat exists.