Soon after the beginning of “All In: Comedy About Love by Simon Rich,” the comedian and actor John Mulaney reads a letter by Mr. Rich, with whom he worked years ago as a writer on “Saturday Night Live,” explaining the mission of this 90-minute piece. “Deep down it’s meant to tell one simple story,” Mr. Mulaney relays, “about how the most important part of life is who we share it with.”

If that description suggests a sentimental journey, think again. Mr. Rich — the younger son of Frank Rich, the former New York Times theater critic and columnist, who in recent years has served as executive producer on hit TV series such as “Veep” and “Succession” — has collected numerous credits as an author and screenwriter himself, among them short stories published in the New Yorker.