Let’s say I had the same brief - keep the name, make a reboot rather than a spinoff.
Seemingly nice high school in a seemingly nice college town.
Buffy is, initially, not very secure about the whole slayer deal.
I’d have her come in to her power in the pilot. It doesn’t have to be a full-on origin story, it could be a quick flashback bit, but I’d show the Chosen One being chosen. By the end of the pilot, she’s ready for the battles ahead, though she still has concerns.
A few friends - we can call them Xander, Willow, and Giles, or not, depending on how close they are. Maybe give possibly-Sunnydale High more recurring characters as well.
The slayer and vampire starcrossed forbidden love thing is so perfect the show did it twice, so I’d need to find a way to spin it. Girl meets boy, boy becomes vamp? But I also wouldn’t want to muddy the waters of let’s kill these soulless monsters on a weekly basis. Hrm. Some unique mystical whatsit to explain his uniqueness, maybe it’s the result of an attack by an ancient proto-vampire, or she tried to stem it off with a transfusion of her blood, which obviously she won’t be wiling or able to for every vamp around.
Decide just how helpful and useful the Watchers are - stick it at not very and keep it there. I say this as someone who ran with the Watchers having a good education programme, alchemists, demonic contacts, a black ops team and more. The slayer getting assigned one whole Watcher should be a big deal for them. (And see what the Charmed reboot is doing with its tweedy mentor guy who is, I kid you not, William from The Watch House. Because I swiped him from Hellboy as I know how to typecast)