When Boris Johnson was foreign secretary he allegedly asked his staff about Leo Varadkar: “Why isn’t he called Murphy like all the rest of them?”

It was presumably a joke – and not greatly appreciated when the leaked remark reached Dublin. But Johnson had hit on a basic truth about Varadkar. He is not a typical Irish politician.

The gay son of an Indian immigrant, a trained medical doctor, and socially awkward, the taoiseach does not fit the usual mould.

Instead of gladhanding constituents and attending funerals, two traditional elements of Ireland’s parish pump politics, Varadkar prefers to read policy papers and strategise with aides. He is, in other words, that person mocked by Johnson: a swot.

Yet when the two met in north-west England on Thursday, in private, away from journalists and the usual panoply of summitry, Johnson was forced to look up to Varadkar, both literally and metaphorically.

The taoiseach is 1.93m ( 6ft 4in) and stands on the shoulders of the European Union – its commission, council, parliament and all 27 member states.

The Irish government’s success in mobilising the EU behind the border backstop has rebalanced Ireland’s historically subservient relationship with Britain.

Yet Varadkar, like Johnson, is under immense pressure. If the UK leaves the EU with no deal it will hammer Ireland’s economy and destabilise Northern Ireland. But accepting a tweaked version of Downing Street’s current plan, or some other plan that weakens the backstop, could torpedo Varadkar’s hopes of winning Ireland’s looming election.

The taoiseach would have got straight down to business.

“He doesn’t have a lot of time for, or interest in, small talk,” said Brendan O’Shea, a doctor who trained Varadkar as a medical student. “We have someone who is forensic about what he thinks should be done. The calculating machine will make a decision about what is the best decision then the politician will switch on and figure out a way to sell that.”

Growing up as a GP’s son in a middle-class Dublin suburb, young Leo declared, at the age of eight, an ambition to become health minister.

A biography says that, as a teenager on a school trip to Northern Ireland, the future taoiseach smuggled fireworks back into the republic, which some might consider a metaphor for his role in Brexit.

Varadkar studied medicine at Trinity College Dublin and joined Fine Gael, a centre-right party. Outspoken about tax cuts and welfare reform, some called him “Tory boy”. His heroes included Otto von Bismarck and Ryanair’s Michael O’Leary.

As a Teachta Dála (member of the Irish parliament) and minister, Varadkar earned a reputation for energy, direct language and social stiffness – some public interactions left aides cringing.

Johnson’s bonhomie, however, may not be totally wasted on Varadkar, who has loosened up since coming out as gay in 2015.

“It’s not something that defines me,” the then health minister told RTÉ. “I’m not a half-Indian politician, or a doctor politician or a gay politician for that matter. It’s just part of who I am.”

It was a brave admission – Ireland had yet to legalise same-sex marriage. The response was overwhelmingly positive and Varadkar went on to succeed Enda Kenny as leader of Fine Gael and taoiseach in 2017, and in the process to go from Mars bar binger to fitness fanatic.

“Coming out changes everybody,” said Tiernan Brady, an LGBT activist who advised Varadkar. “In accepting who you are to yourself, and that you can still go on to lead your party and your country, that’s been really affirming. That brings you a whole level of confidence.”

Some Irish backstop sceptics warn that Varadkar’s government may be suffering from over-confidence, that bending Brussels to Dublin’s agenda may backfire, and that the broad national consensus behind the strategy may deter the taoiseach from compromising.

“Varadkar’s personal political interests have diverted from the national interest,” said Dan O’Brien, the chief economist at the Institute of International and European Affairs. “His political career could be ended by making a concession on the backstop. Fianna Fáil and Sinn Féin would come down on him like a tonne of bricks. It’s never a good idea for a leader to be painted into a corner.”