Brian Cox had no intention of tackling a meaty character like Lyndon Baines Johnson in Robert Schenkkan’s “The Great Society.” He’d moved away from big theater roles to focus on parts in movies and television, most prominently HBO’s “Succession,” on which he plays Logan Roy, the founder of a media empire.

“At my age,” says Cox, 73, “I should be watching the daisies grow.”

He signed on to do a staged reading of “The Great Society” for “a nice fee,” but it went so well, director Bill Rauch approached him about doing a full-scale production.

Cox hadn’t played such a gigantic role since he did King Lear, nearly 30 years ago. Though terrified, he said yes, and hired an old dresser friend to “beat the lines into me.” And then something “bizarre” happened: “My old self, the young man who could do these parts, sat on my shoulder and said, ‘You’ll be all right.’ And the muscle memory came back. I thought it had gone, but it had not.”

The first previews were a little rocky. Cox says he wore an earpiece in the beginning, through which he was fed lines. “My brain wasn’t moving fast enough,” he says. “The trick is speed. Johnson spoke very fast.” Now, several weeks in, he’s beginning to enjoy himself.

“The Great Society,” which opened last week at the Vivian Beaumont, follows Johnson’s drive to pass the Voting Rights Act of 1964 while juggling the growing Vietnam War that eventually brings down his presidency. Asked if he spoke with Bryan Cranston, who won a Tony playing LBJ in Schenkkan’s earlier play, “All the Way,” Cox flashes a menacing look. (He’s a very good villain, having played the first Hannibal Lecter in 1986’s “Manhunter.”)

“You’re joking, right?” he says. “‘Oh, hello, Bryan. I’m about to play a part you played and believe you were very successful in …’ No way. We’re different animals.”

Cranston wore prosthetic earlobes in the role. “I’ve already got those,” says Cox, tugging on one. “My problem is height. So I wear lifts to get myself up to 6 feet. Higher than that and I get nosebleeds.”

Cox has been in the business since he was 15, but is only now finding fame as Logan Roy. He was doing an interview at MSNBC the other day and says two top executives there chased him down the hall to tell him they were binge-watching “Succession,” now in its second season.

“They said they had to meet Logan Roy,” he says. “These two powerful men were like little boys.”

When he was offered the part, he says, Logan was supposed to die at the end of the first season. “So it’s one season only,” said Cox, after a long discussion with a group including creator and showrunner Jesse Armstrong and executive producer and director Adam McKay. There was a pause and then Armstrong said, “I don’t think so.”

Cox sees similarities between Logan and LBJ.

“They’re both self-made,” he says. “And they have all the problems that come with that. They have nothing to adhere to. They create their own value system, and when they get enough bruises, their value system becomes corrupted. What I love about Logan is that he is mysterious. We can never know him.”

One thing Cox is adamant about: Logan “loves his children deeply,” even though he often undermines them. “He knows the situation they are in, and he knows he is responsible. But they have to own up to who they are. That’s the whole point about ‘Succession’ — owning who you are. And his children come up short every time because they can’t do that.”

As for his burgeoning fame from the show, Cox says: “It’s weird. I’ve seen people come and go. I’ve seen what this profession does to people. How cruel and unforgiving it is. But a wonderful director, Michael Elliott, said to me in the early 1980s, ‘You’re going to be an actor whose time will come.’ So I’ve always known it was the long game. I knew if I stuck around, and had the courage for reinvention, there would be satisfaction.”

Born and raised in Scotland and now living in Brooklyn, Cox has played many great roles, including an unforgettable (and very bloody) “Titus Andronicus” I saw at the Royal Shakespeare Company in London in 1988. He’s also one of the few actors left who worked with the three giants of the British theater: Ralph Richardson, John Gielgud and Laurence Olivier. He does hilarious imitations of all three.

“Ralph was the most eccentric man I ever met,” Cox says. “One day he said, ‘I’m not going to rehearse today. I’m going to hide.’ And he put his hands over his face. The director came in and said, ‘Ralph, what’s going on?’ And he sat there and said, ‘I’m hiding.’

“Gielgud sparkled with wit. He hated the National Theater, a ’60s concrete building along the Thames. When we were doing ‘Julius Caesar’ there, he used to say, ‘I keep waiting for my flight to be called.’

“Olivier was ‘the silver fox.’ At 80 and frail, he played King Lear (Cox was Burgundy). ‘I’m so tired,’ Olivier said one day. ‘So tired. I don’t know my lines.’ The great actor Michael Hordern had played Lear the night before on the BBC. ‘Did anybody see Michael Hordern last night? He knew his lines. I don’t know my lines.’ There was a pause and then a roar: ‘But I’m still a better f–king actor than he is any day!’

“We’ll never see their likes again,” says Cox, “and how incredibly lucky I was to work with all three.”

“Len Berman and Michael Riedel in the Morning” airs weekdays on WOR radio 710.