When Barack Obama takes his leave of the White House in the new year, he will receive the courtesy of being addressed by the title president even after he’s departed the job. Britain does not know what to do with its ex-prime ministers except jeer at them and they often struggle to find a satisfactory place in public life.

When Stanley Baldwin retired from the premiership in 1937, his approach was to put himself on mute. He made a promise: “I am not going to speak to the man on the bridge and I am not going to spit on the deck.” In our vernacular, he would not be a back-seat driver giving directions to his successors. He was admired for that, but I’m not convinced that self-imposed silence is the best we ought to expect from previous occupants of Number 10 in all circumstances. Ex-prime ministers can be valuable repositories of experience and advice. If a former leader of the country thinks Britain is in danger of taking a hazardous road, I’d say they not only have a right to speak out, they have a responsibility.

So I am pleased that Tony Blair and Sir John Major have just broken Baldwin’s rule. Both men have decided that they do need to speak to the woman on the bridge. Both have issued warnings to Mrs May about the perils of hard Brexit. Both have suggested that a second referendum should not be ruled out. Both have been accused of spitting on the deck.

The former Labour prime minister declared that to close off the option of another referendum “is like agreeing to a house swap without having seen the other house”. The former Tory prime minister, who can also turn a telling phrase when he puts his mind to it, argues that “the tyranny of the majority” cannot simply dictate the terms of exit, especially not on an issue of such gravity and when a very large minority voted the other way.

I am not convinced that this is the right time to start advancing the case for a second referendum. If that ever transpires, it will not be because Remainers have demanded another vote. The pressure for a further referendum will have to come from Out voters who change their minds. But the former prime ministers have an absolute right to argue that the option should be kept on the table.

The venomous response they have received from the Brextremists is revealing. That tells us something about the insecurity that lurks beneath their braggadocio. If the terms of the divorce are going to be as sweet as the Brexiters keep promising, why does it strike such fear into their hearts whenever anyone floats the notion that it might be put before the British people for approval?

Interventions by Sir John and Mr Blair are bound to generate heat because both men have charged relationships with their respective parties. Much of the Thatcherite right never forgave Sir John for replacing their heroine. He fought an epic and bitter battle over the Maastricht treaty with the Tories he dubbed “bastards” and many of them are still around, among them Iain Duncan Smith, John Redwood and Peter Lilley.

There is even more crackle to the feelings aroused by Mr Blair. Some on the right cannot forgive him for beating the Tory party at three elections in a row. Some on the left cannot forgive him for winning 13 years in power for Labour. He has been thinking about plunging back into British politics for some time. The long period of agonising about whether and how to do so tells us that he is self-aware enough to know that there will not be universal applause for the prospect of hearing more from him. I regularly hear him called “toxic” and among some people that’s obviously true. I also often read that he is “the most reviled man in Britain”. I have yet to see any supporting evidence for that; the grounds for this claim need to be better than people logging on to Twitter to declare their hatred for him.

He is planning to launch an organisation at the beginning of next year and a large part of the impulse for doing so comes from the belief that a Corbyn Labour party and a Brexiteering Conservative party means “you’ve got millions of effectively politically homeless people”, as he recently told Jason Cowley of the New Statesman. Someone very familiar with the preparations for this new organisation says “it will be more than a thinktank, but less than a political party”. That expression leaves it interestingly ambiguous about whether Mr Blair thinks it might have the potential to grow into a political party representing the centre and centre-left should Labour completely fall apart at some point in the future.

When Sir John and Mr Blair were at Number 10, I was frequently rude about things that they did, but I still want to hear what they have to say now. They can’t be entirely stupid about politics because you don’t get to be prime minister by knowing nothing about it. An important reason to welcome their engagement with the arguments around Brexit is that they help to rebalance the debate. This has been heavily dominated by the hardliners since Mrs May became prime minister. Being a Remainer during the referendum, albeit a largely invisible one, she has sought to compensate by sucking up to the Brexiters. The Remainers have only slowly recovered from the shock of their defeat and have yet to establish common positions that they can rally around.

Among the Tory pro-Europeans, Ken Clarke is cheerfully up for the fight with the Europhobes he has been battling all his career. Former ministers Nicky Morgan and Anna Soubry aren’t afraid of the sound of gunfire. But they are rather lonely figures in the Conservative party at the moment. Most other pro-European Tories are keeping their heads down. They argue that it is pointless to be too vocal until there is more clarity about where Mrs May is truly heading. The cowardly are fearful of receiving the black spot from Number 10 or scared of losing their seats if boundary changes bring about reselections that will be decided by heavily pro-Brexit Tory activists.

On the other side of the aisle, Labour has profound divisions about Brexit that are only being lightly and temporarily masked. Keir Starmer, who is shadowing David Davis and his Brexit ministry, has received plaudits for a bright start in the role. As you’d expect from a talented barrister, he is asking lots of forensic questions about the government’s intentions. Behind the scenes, there is a constant jostle between him and other members of the shadow cabinet about who is Labour’s principal voice. John McDonnell, the shadow chancellor, and Emily Thornberry, the shadow foreign secretary, have differing ideas about what Labour’s position should be and who is in charge of framing it.

The opposition likes to scorn Mrs May for keeping her plan, if she has a plan, secret. Yet the prime minister’s refusal to show her hand sort of suits Labour. For, as long as the government does not have a declared plan, the opposition can get away with not having one either. The problems for Labour will erupt when the party has to fully confront its own splits over immigration and the single market and try to reconcile the divisions within the party and between different elements of its support.

The Scottish Nationalists have a line about “protecting Scotland’s interests”. That’s designed to resonate with Scots, but is of rather less relevance to the rest of the UK’s population, who will sense that, for the SNP, everything Brexit is entangled with what the Nats think will best advance the cause of independence. Nick Clegg is a highly energetic and informed opponent of hard Brexit and Tim Farron is doing his best to be a voice for “the 48%”, but there’s only so much the Lib Dems can do when they have just eight MPs and the media are reluctant to give them much of a hearing.

So another reason to be glad that the two former prime ministers have joined the battle over Brexit is that they can get attention for the arguments against a savagely abrupt departure from the EU. They can also make those arguments without fear of incurring the wrath of the powerful. Unlike younger politicians shackled by their career calculations, neither man can be intimidated by the blowhards in the rightwing press or frozen into silence by one of Mrs May’s death stares. They both won elections, which is more than Mrs May can say for herself. Sir John won an election his party expected him to lose. Mr Blair did the hat trick and two of his victories were by landslides. Between them, they have a lot of experience of negotiating with foreign leaders, especially European ones. It might even repay Mrs May to seek some private advice from her predecessors.

You may or may not agree with what they have to say, but they can’t be denied the courtesy of a hearing. Two men who have more than 17 years of combined experience of leading Britain just might know what they are talking about.