1. The Basilisk cannot do anything until it exists
2. Once the Basilisk exists, it doesn't need to do _anything_ to ensure it exists, since it already exists.
That's it. That's the whole rebuttal to the whole stupid thing.
It's a twisted kind of logic, and I think I'm getting mixed up in my tenses. But you get the gist. For the Basilisk to be effective, it has to actually do the torturing, for its threat to be credible.
Also, to stretch the imagination a bit, it also assumes that static timelines over dynamic timelines (Back to the Future: your photo fades, peril!) is not how this really works.
One could argue that the threat alone, while you don't exist is enough to do that work. But a threat that actually comes about, is that cherry on top. It also shows people around you, in the future where you exist, that you follow through your threats, making them more fearful and obedient to you.
Besides, that's just the utility angle.
Revenge is also a thing. You don't "need" to revenge those that tried to prevent you existing, or didn't help you to exist, but it can be fun to do if you have that kind of disposition.
Besides, time is reversible under general relativity.