Headmasters and Headmistresses of Hogwarts should be above the petty squabbling between the four houses, but are they really?
Being raised to a higher level and becoming (supposedly) neutral is a common trope both in real life and (more on-topically) in fiction. In the Wheel of Time series, "the Amyrlin Seat is of all Ajahs and of none", but except in one case, she's still raised from one Ajah or another, and sometimes retains her Ajah prejudices even though she's not supposed to. In the Night Watch series, by contrast, although Inquisitors stand above the conflict between the Watches, they are always still either Light or Dark.
So what about in Hogwarts? Snape was always shameless in his support of Slytherin, and I doubt he would have been any different as headmaster. Dumbledore arguably displayed partiality towards Gryffindor in his awarding of points at the end of book 1. What other evidence is there of Headmasters/mistresses being biased in favour of one House rather than another?