Question: How did Johnny Carson maintain his cheerful TV persona despite his personal struggles?
It was apparently a facade.
Johnny Carson was a cold individual who didn’t suffer fools gladly and who had few interactions with friends over his lifetime. As was shown with relations with Joan Rivers and Lola Falana (former friends and frequent guest on his program) if you crossed Carson he wasn’t about forgiving or forgetting and you disappeared from his presence.
Even his on-air personality had its limits as you can see if you watch old clips of Rodney Dangerfield, Robin Williams, and a few other guests who for someone reason stretched Carson’s patience to the breaking point. Even though the camera cuts away from Carson quickly in certain shots you can see his pleasant manner drop and even him glaring backstage (apparently at his longtime producer Freddie DeCordova) when a guest was not performing up to snuff or was stealing Carson’s “thunder”.
Carson’s personal struggles were significant with the unexpected death of his son Richard being among the most prominent. However Carson problems were mostly self-inflicted with four marriages, his alienation from other people and in the end his alcoholism, and his heart disease caused by his smoking.