No, the patriarchy existing is not bio-essentialism

I wasn't intending to claim that — the patriarchy can and should be overthrown. The patriarchy harms people of all genders.

I was thinking of a particular line of reasoning following from it to justify men deserving hate (for instance, "men deserve hate on account of being men, because they are in a system where they are privileged") might be though.

I guess you're right that a better line of reasoning could follow, e.g. men, exploiting this position of privilege and thus inflicting harm, deserve hate as a result.