Changes to Hogwarts After the War

As best I can tell, Dumbledore was Headmaster of Hogwarts for somewhat over forty years.1 We are unsure how long he was Deputy Headmaster before that, and in what state he "inherited" control of the school. Irregardless, the consensus is that changes were necessary after the war. There is much less consensus on what those changes look like, how successfully they were implemented, and by whom.

The available secondary information on her strongly suggests that she was appointed Headmistress in the immediate aftermath of the final battle, and remained in that role for at least the next twenty-one years.2

What kind of Headmistress was she? Did she seek to perpetuate Dumbledore's policies?3 Can she crack down on discipline without changing the very nature of Hogwarts?4 In either case, I do not think that she would support disbanding the House structure entirely, nor Slytherin house specifically. In the first place, I do not think that there is reason to disband the House structure, as I imply as I look at Hogwarts, Houses seem to be a staple part of life at most boarding secondary schools across the United Kingdom. Given you are going to have Houses at all, lets be proud of our tradition and keep our Houses named for Founders (as that too seems common), but reform relationships and the nature of the sorting itself.

Slytherin is to be the house of ambition and cunning. Neither of those is a trait restricted to those of one lineage. If we get rid of legacy house placements, and install a zero tolerance for prejudice and bullying, we can actually sort people where they belong. Do Crabbe or Goyle show any ambition or cunning for example? Just by breaking up "traditional" placements we can probably do much to reform the relationships between the houses.