As others have mentioned, BP itself is no issue to the site culture. And even if you want to drill into the "problematic" threads, like the hate one, it's doing nothing to site culture by itself.
If anything, it's a reflection of how poorly adjusted some people are here. And they did not need these threads to think this way.
The site is seemingly full of incels on both sides of aisle. And, despite being so alike in their blind reciprocal hatred, they just can't help but to poke the other camp any given occasion or seek for examples to validate their opinion.
I would imagine that most people who do not harbor such extreme feelings towards the other sex have long given up on trying to reason with either camp, if they ever even tried to.
I think that before the man hate thread was even a thing, it was pretty common for a segment of users who now exist in the woman hate thread to openly shit on women site wide.
Somehow, it only becomes an issue now that it's no longer a one way street.
I don't want to read about any side of the argument, but it does not seem objective to blame one side more than other to me. If anything, I am glad they have containment zones and wish they stay there as much as possible.