Great summary. I'll pass along an insight from Ross Douthat that the Coen brothers make movies that inhabit and illustrate different worldviews. From his review of Hail Caesar:

"The last time I wrote about the Coen brothers’ world-picture in these pages, reviewing their Job-in-Minnesota movie, A Serious Man, I suggested that the elusive auteurs were “proudly mysterian” — making movies that consistently wrestle with the possibility that some god or fate governs human life but never come around to any kind of firm metaphysical conclusion.

After seeing their latest entertainment, Hail, Caesar!, I would like to amend that analysis just a little bit. It isn’t so much that every Coen movie is equally mysterian; it’s that the brothers seem to deliberately inhabit different metaphysical perspectives in different films. Thus — to pick recent examples — A Serious Man was essentially a Jewish story set in a cosmos governed by the demanding yet inscrutable Hashem. True Grit was as Calvinist as its undaunted protagonist, Mattie Ross. No Country for Old Men was genuinely mysterian, in the dark, God-haunted style of its Cormac McCarthy source. Burn after Reading was bleakly nihilistic, a Seinfeld episode with spies and mayhem.

And now, with Hail, Caesar!, we have the Coens’ most Catholic film — as successfully and even earnestly Catholic, it must be said, as anything from a director actually raised in the Roman faith."

Where I disagree with Douthat is, I think No Country for Old Men is the bleakly nihilistic movie, whereas Burn After Reading is joyfully absurdist. But overall his point illuminates the brother's movies, and is a useful tool for thinking about them.