The one concept from Nietzsche I see everywhere around me in the world is ressentiment. I think much of the master-slave morality stuff was too specific and now feels dated 130 years later, but ressentiment is the important core that's still true and going to stay with us for a while; it's like a powerful drug that won't let humanity go. Ideological convictions and interactions, myths and movements, all tied up with ressentiment or even entirely based on it. And you're right, I would have everyone read Nietzsche - not for practical advice or predictions, but to be able, hopefully, to understand and detect this illness in others and especially oneself.