Hate is a strong emotion and requires work to maintain.

Hate is just disappointed love. You can't hate what you are indifferent to. The true opposite of love is indifference, not hatred.

Love / hate requires an object, and by strict definition, not just any object, but a sentient being with admired character and behavior traits that you love or want to possess or have access to in some way. So I suggest it's not possible to hate a concept (other than metaphorically). You can dislike it, but not really hate it. In fact to my recollection until this moment I've never even heard anyone posit that "hatred of hate" is somehow even an issue. Is it really? I don't know. I kind of doubt it.

Some feelings are "negative" (unpleasant) but this doesn't constitute "negativity" or a bad attitude necessarily. When I have grieved the passing of a loved one for example, that is unpleasant but also perfectly natural and trust me, the last thing you want to hear in that situation is someone claiming you are "being negative". We should not confuse unpleasant feelings with dysfunctional or useless feelings. We should not automatically and reflexively push away or avoid negative feelings in ourselves or others. Feelings (pleasant and otherwise) are just information (often with high signal-to-noise ratio, but information nonetheless).

So. While I also (probably for diffferent reasons) disagree with the typical fundamentalist "hate the sin, love the sinner" ethos, I do not regard a strong distaste for hateful, otherizing or other bad behavior to be inherently problematic, either. Otherwise how would we be motivated to change for ourselves and others. How would we sanction agreed societal harms. How would we avoid enabling hurtful behaviors?

I don't believe anyone who tells me they do not hate something.
Hate is a natural human emotion. I don't believe in denying emotions.
What's important is how they're expressed.
There is no positive without negative. Balance must be maintained.

I'm okay with hating hate. I certainly wouldn't want to love hate or even tolerate hate.

There has to be some hate:

For mankind to ascend to its higher form there must always be some sense of danger, inequality and self-love. Without the aforementioned danger and striving for life, we can never ascend and will only be fit to belong to the 'herd'

We need danger, we need to strive for life!

I suppose another way to put it is “It hurts me deeply to see all of this hate in the world.”

Indeed it is.