I'm a little sad that Snape is terrible here. Though it does fit his canon personality quite well. I suppose I've just been reading too much alright-Snape fanfic to like his assholishness here.
I think that it was not really bad, considering that he has been through enough to damage him socially, and that the child he is dealing with is from "the family he couldn't have" and whatnot. A child came barging into his space and he wanted that child gone... I think that people have a right to their space.
This of course all gets thrown right out of the door the moment that one realises that he is a school teacher. He should never have been anywhere near to a position that requires interaction with children, especially not this specific child. This is, at best, a massive failure as a teacher, and could potentially be career-ending depending upon the local standards. The locality is Hogwarts, and the standards are... well, they had the big bad on-staff, after they were declared the big bad. Now, the big bad was in hiding, but, well, when they have some of the most capable wizards, and most powerful artefacts, one might think that they could do a basic scan for "loci of corruptive malevolence" once in a while... and there was also Ubmrage, and a literally cursed posting that can't have been getting the highest bids... and the ghosts of former students who died violently on-campus is not exactly a great look...
Like, this level of automatic hostility towards a student is pretty bad, but seems outright civil by the standards of the time and place. I mean, she wasn't even accosted by a large hungry carnivore, or deliberately exposed to her worst fears with perilously little preparation, or sent dozens of metres into thin air on a dodgy broom that never sees maintenance...
I think that he better fits as a university professor...
It is also worth a thought that this may have been a really bad introduction. This was a student who was clearly failing to be a good little sheep sticking with the herd, and thus was probably one of those "creative" types who are always causing trouble.