I don't really think there was anything to "get". It wasn't a particularly subtle satire.
I didn't know who the fuck Ariana Grande or Kid Kudi were until my daughter told me who they were so at least now I feel my indifference is justified. So while I enjoyed the movie, I think I still probably missed tons of inside jokes.
I take a bit of offense that it's been compared at all to Dr. Strangelove because come on! I'm thinking it terms of black comedy satire, something like In the Loop or even Burn After Reading did a better job of capturing political rot.
I think this movie excelled because it was really tight and captured some vibes from lots of genres. But in the end, it's almost as if it was almost "too soon" in relying on the idiocy of the past four years. Too many of the characters and jokes seemed like copies of real idiots.
Adam McKay arguably did my favorite movie of the last five years, The Big Short. I think this one fits in nicely between his Will Ferrell comedies and something as cool and great as The Big Short.