I adored the Young Pope - and enjoyed the first episode of the New Pope. Yet this one was weak. With so much self-conscious quirkiness, this episode entered The Life Aquatic with Steve Zissou territory, though without the payoff.
Malkovich, while a wonderful actor, is misplaced in this role. He's utterly unconvincing as an upper-class Englishman, sane or not. His accent is obviously fake and grating, recalling Tyrion Lannister's voice (another comically-bad English accent by an American actor). As the son of a duke, he would be Lord John Brannox, not Sir John Brannox - though in reality he is John, Cardinal Brannox. "Sir John" refers to a baronet or knight not entitled to a higher style. Since he is apparently the only son of this duke, he'd more likely have a courtesy title (likely in the grade of marquess or earl, though not exclusively those), and be known as Lord Titlename. His parents, meanwhile, would be the Duke and Duchess of some placename, not Duke and Duchess Brannox. This is not arcane knowledge and could be learned in a quick web search, so there is no valid excuse for the writers being ignorant of such vital knowledge which would have lent much-needed authenticity to what appears to be a new lead character.
"Sir John" is dressed like an Edwardian aristocrat without explanation despite being a very senior churchman. In 2020, this would be well beyond a mere eccentricity, and worthy not only of constant comment, but sectioning. This dress alone would disqualify him out of hand for consideration as head of any responsible enterprise, and certainly as head of the RC Church. Laughable that the Vatican Secretary of State would come cap-in-hand to beg such a loon to "accept" the Papacy, especially on the pretense that he personifies moderation.
We can only hope that most of this episode--at least all of it involving the ludicrous visit to the country house--is part of Lenny's fevered dream sequence, and not a portrayal of canonical events in the serial.