The prime minister needs help. No, that's not a plea for Gordon Brown to undergo a course in anger management – not even one laid on by the self-styled National Bullying Helpline, which inserted itself so clumsily into public life this week. Instead, it is a request to look less at Brown – even the fist-flying CGI version that's fast become a viral smash – than at the office he fills. For what the row over threatened punches and cowering aides has revealed is that the job of prime minister has become all but ­impossible for a mere human being to do. It is, as one senior Whitehall figure once put it, "beyond any mortal".

We have long insisted that the ­occupant of No 10 be firm, decisive and courageous. Brown was pilloried ­mercilessly when he "bottled" that phantom election in 2007. But, it's now clear, we also want someone gentle, polite and emotionally intelligent. We insist those at the top maintain the dignity of their office – a past Labour leader was ridiculed for wearing the wrong kind of coat at the Cenotaph – but can also exchange banter like a regular guy, so that Brown had to chuckle along chummily as Piers Morgan asked whether he was a "plonker".

We insist on courtesy, but when we get it, it's not enough. John Major's manners were impeccable – apparently the typists in Downing Street's garden room liked him – but he became a national joke who was ejected from office in the largest landslide since 1935. The "Garden Girls" were fond of Tony Blair, too, and he could be charming on any daytime TV sofa. But plenty of Britons would have preferred a ruder, coarser Blair – if that meant he didn't lead the country into a disastrous war.

We want a prime minister who is simultaneously steel-willed and flexible, statesman and good bloke, authoritative dad and nurturing mum. No single person could ever meet all those expectations. "The job requires the energy of Gladstone, the flair of Disraeli, the balls of Lloyd George, the administrative gifts of Attlee, the style of Macmillan and the sleeping patterns of Thatcher," says the scholar Peter ­Hennessy, adding "human beings don't come like that".

In part, this is not about personalities but about the system. The role of prime minister is already enormous: he wields more centralised power than, for example, an American president – and more is piled on his plate every day. In Hennessy's book on the job, one chapter, tellingly titled "The stretching of the premiership", lists his basic duties: hiring and firing ministers, chairing the cabinet and its most important ­committees, controlling the civil service, maintaining relationships with heads of foreign governments, close management of the country's foreign and defence policy, collaborating with the chancellor in budget-making, co-ordination of the security services, and supervision of the government's political strategy. Oh, and a prime minister must also be a party leader. Does that sound to you like a job for one person? Or should only super­heroes apply?

Such a job description asks the impossible. A panel of US wise men came to a similar conclusion when they examined America's highest office in 1937. The Brownlow commission concluded pithily that, "the president needs help", proposing a heavily reorganised and reinforced executive office. The man they were rescuing was none other than Franklin Roosevelt. In other words, even FDR, one of the presidential greats, was overwhelmed.

But now there is an additional layer of expectation – and it comes from our ever more voracious media, demanding that the national leader appear before a camera or microphone around the clock, seven days a week. They must get the balance right between gravitas and warmth, adjusting it for the audience, the format, the moment – judging each performance as adroitly as the most seasoned actor. The same scholars who rate Clement Attlee as the most skilful postwar British prime minister agree this bald, modest man would not have been selected as so much as a parliamentary candidate – let alone have reached Downing Street – if he had had to survive in today's conditions. As for FDR, it is hard to see how an aristocratic New Yorker in a wheelchair would have made it in the television age.

Part of the pressure is intensely practical. A first-rank politician has to be ready to appear on Newsnight one night, then pop up bright and early on the Today programme the next morning, managing to stay word-perfect and error-free, and yet also sound warm and human. Somehow, in the six hours in between, they have to maintain a normal, wholesome family life – otherwise they will be branded a weirdo. We slam Brown for looking rumpled and exhausted, but woe betide him if he makes the mistake of being on holiday when a major story breaks. "Why is the prime minister ducking the issue?" we will ask.

Small wonder that several PMs have been deemed to have lost their marbles. Some observers reckon it's the pressure of the job that does it; others suggest it takes an extreme type to seek the job in the first place. Psychologist Oliver James says he has discerned the signs of ­narcissistic personality disorder in the current and last incumbent, defined as "an exaggerated, grandiose sense of their own importance".

Is there anything that can be done, or should David Cameron make an advance booking with the shrink in preparation for the day it all becomes too much? David Owen, a rare example of a medical doctor who served in the cabinet, has several remedies to offer. First, he is dismissive of bleatings about media pressure: he suggests prime ministers simply learn to say no. (He reckons Thatcher would have gone nowhere near Piers Morgan.) That goes for much of the workload. If he is invited to 10 meetings of the European council, then go to four – send the foreign secretary to the rest. In other words, learn from top-flight CEOs – also doing jobs many regard as impossible – and delegate.

Another tip: term limits. Prime ministers should be allowed to serve no more than two fixed, four-year parliamentary terms. The timing makes sense because eight years tends to be the moment when what Dr Owen diagnoses as "hubris syndrome" kicks in: better to get out first.

But the most significant change should be to our system of government. Some are tempted to go the whole hog and make explicit what has already ­happened, recasting the premiership as a directly elected presidency. But that too would be an impossible job. A more direct solution for the overworked PM would be a return to cabinet government, the primus spreading the load among his pares.

Easier said than done, though a coalition government would force a prime minister to share power. If it's a single-party government, then it comes down to the character of the person in the job. Cabinet government is possible if the PM genuinely wants to hear collective advice, rather than relying solely on his own instincts. Such a leader wouldn't mind being off the air, allowing others to take the strain; he might even have other interests besides politics. That might sound fanciful, but Cameron and Brown had better start thinking about it. Because right now, they are competing for a job neither of them – and no one else – can do.