It used to be argued that British drama is driven by a fascination with public affairs and its American counterpart by a preoccupation with private lives. On the evidence of a week’s intensive theatregoing in New York, I would suggest that hoary generalisation has been blown to smithereens. At a time of potential impeachment, political polarisation and profound uncertainty, American theatre seems to be heavily engaged with the wider world.

The most surprising – and controversial – new play I encountered was Will Arbery’s Heroes of the Fourth Turning at Playwrights Horizons. It offers a direct rebuttal to the stock charge that modern drama simply reflects the liberal assumptions of the theatregoing audience. Arbery writes about a group of deeply conservative Catholic friends who meet in Wyoming for the installation of a new president at their old college. The play neither caricatures nor cauterises these figures. It shows them to be passionate, articulate and as capable of quoting Hannah Arendt as Heidegger. At the same time, it suggests they are just as divided as any other section of modern American society.

What is startling is their vehemence. The most extreme figure is Teresa who admires Steve Bannon and is excited by the idea of a war to save western civilisation. Her ex-lover, Justin, sees proximity to LGBT people as a threat to Christian children and advocates training in marksmanship for the college’s students. But Arbery also presents us with the alcoholic Kevin who feels spiritually empty, and the more rapturous Emily who worked for an anti-abortion women’s organisation helping domestic abuse victims. Everyone, it seems, loves Emily except her mother, the newly elected president, who turns up at this outdoor drinks party and argues for a return to religious orthodoxy.

Startling vehemence … Heroes of the Fourth Turning. Photograph: Joan Marcus

If anything unites these squabbling Catholics, it is their dislike of President Trump who is variously described as “a gross monster” and “chemotherapy”. But Arbery’s main point is that there seems to be a malaise affecting the whole of the United States of which these fervent rightwing Catholics are yet one more symptom. Arbery overdoes the symbolism – as with a noisily explosive generator that echoes the breaking string in Chekhov’s The Cherry Orchard. But his play is robustly challenging and Danya Taymor’s production boasts a fine Edward Hopper-like set from Laura Jellinek and striking performances from Zoë Winters as the poster girl for the new apocalypse and Julia McDermott as the sickly Emily.

If Arbery’s play offers little in the way of comfort, Soft Power at the Public Theater addresses the nation’s ills with the breezy bounce you expect in a musical. It has a strong pedigree in that the play and lyrics are by David Henry Hwang (M Butterfly) and the score by Jeanine Tesori (Caroline, or Change). The framework is complicated and derives from a street stabbing in which Hwang was the victim. But what we see is a fantasy musical that he imagines in his fevered post-op dreams and that offers a parodic inversion of The King and I. Instead of a British governess lecturing a Siamese monarch we see a Chinese film producer, Xue Xing, arriving in America and offering a lesson in civics and good governance to Hillary Clinton.

Breezy bounce … the musical Soft Power. Photograph: Joan Marcus

There are some genuinely funny numbers, including one where Xue tries to teach the stiff-lipped Clinton (neatly played by Alyse Alan Louis) how to pronounce his name. As the all-Asian chorus line, bewigged to resemble standard Broadway hoofers, strut their stuff, we realise just how offensive it is when, as more often happens, occidental actors play oriental roles. But the virtue of the show is that it pins down the current American crisis with jaunty irony. “The ballot box has thrown your country into chaos,” claims the communist Xue. While Clinton agrees, she plangently sings that you have to stick with democracy. She even finally endorses a Chinese proverb that proclaims: “Good fortune will follow if we somehow survive.”

While Hwang’s musical and Arbery’s play deal with the current crisis, Robert Schenkkan’s The Great Society at the Vivian Beaumont looks back to the historic past: specifically to the last years of Lyndon Johnson’s presidency from 1965 to 1968. What is immediately striking, in sharp contrast to today, is the moral purpose that drove LBJ and his success in getting, with the help of Republicans in the Senate, both the Civil Rights Act and the Voting Rights Act on to the statute book.

But, as shown by Christopher Hampton’s 2013 play Appomattox – which has sadly never been seen in Britain – LBJ was one of the great tragic heroes of modern times. A progressive politician and genuine idealist, who got 104 bills through Congress, will always be defined by his disastrous escalation of the war in Vietnam. Brian Cox, who plays Johnson in Bill Rauch’s energetic production, doesn’t look much like the man himself but it is still a titanic performance that captures LBJ’s political guile and ingenuity as well as his descent from a great height.

Odd relationship … Mary-Louise Parker in The Sound Inside. Photograph: Jeremy Daniel

Not all New York plays are overtly political. I caught an intriguing work by Adam Rapp, The Sound Inside, about the odd relationship between Bella, a 53-year-old professor of creative writing at Yale, and a young student, Christopher, who has a passion for Dostoyevsky. Rapp keeps you on tenterhooks as to the real reason for the couple’s extra-curricular friendship and, in a highly literary play, makes a number of shrewd points. One is that fiction is, more often than not, camouflaged autobiography. Another is that, in today’s world, readers (and theatre audiences) crave elliptical conclusions.

Mary-Louise Parker is brilliant as the self-contained, solitary Bella and Will Hochman suggests a dark undertow to her inquisitive student. I only hope that if David Cromer’s production makes it to London, as it surely will, it finds a more intimate space than Broadway’s Studio 54. But this is a fine play that, even if not directly about public affairs, proves that uncertainty is the defining condition of modern life.