The thing is that belonging to house hufflepuff isn't supposed to be some arbitrary selection but is supposed to be about innate, important qualities about the members of such house which is what it takes to qualify. We also know that the Hufflepuff blessing gaves us, or depending on how it works the choice meant Harry naturally had this quality enough for the blessing to apply to him, a absolutely absurd amount of dedication and ability to power through hardship in pursuit of his goals.
My suspicion is that much like the example I gave, the blessing allows people in house hufflepuff to pick up on these qualities, and that the sort of people who are able to get picked by house hufflepuff are the kind who would strongly respect those aspects and see them as things that mark someone as leadership material. That it becomes stronger if harry is part of house hufflepuff isn't suprising, considering one of the traits that hufflepuff epitomizes is loyalty and cooperation among house members to a even more thorough degree in this AU per Birdsie, so it makes perfect sense why the sort of people who'd get picked to be part of the house would show more deference to someone in the house itself with those qualities.
Does that make sense?