The Queen’s first prime minister was Winston Churchill. Her 14th is a Winston Churchill tribute act. So perhaps it’s no surprise that Boris Johnson’s debut performance as prime minister included a sub-Churchillian riff, extolling Britain’s readiness not to fight the Nazi menace but to cope with a no-deal exit from the European Union. “The ports will be ready and the banks will be ready and the factories will be ready and business will be ready and the hospitals will be ready,” he said, until one half-expected him to declare that we shall be ready on the beaches, we shall be ready on the landing grounds, we shall be ready in the fields and in the streets and in the hills.

But if Johnson has always wanted to be Churchill, the address he delivered on the steps of Downing Street and his first exercise of raw prime ministerial power suggest he wants to be two other people too. Throughout his six-week campaign for the Conservative leadership, friends and enemies alike wondered which Johnson would prevail: would it be the twice-elected mayor of London, who won over a Labour-voting, progressive city, or would it be the leader of the Vote Leave campaign, whose lies and insults have led the US president himself to hail him as the British Trump? The complexion of his first cabinet suggests victory for the latter. But there are also signs that he wants to be both, that – not for the first time – he wants to have his cake and eat it.

Start with his brutal purge of more than a dozen cabinet ministers, most of them former remainers. That was the handiwork of the Brexiteer-in-chief, installing his own faction into all the key posts so that leavers’ control over – and therefore responsibility for – Brexit will now be total. There will be no one else to blame, and no place to hide. This is a Vote Leave government now. As Johnson put it: “Never mind the backstop – the buck stops here.” (Not that Brexit explains every blade wielded in this extraordinary night of the long knives: leavers Liam Fox and Penny Mordaunt were sacked too, apparently for the crime of backing Jeremy Hunt, suggesting Johnson has a mafia don’s view of disloyalty.)

The result is a top table that is startlingly right wing. Dominic Raab as foreign secretary, along with the pro-death penalty Priti Patel at the Home Office, puts two people once considered on the wilder Tory fringes at the heart of government – not to mention Jacob Rees-Mogg attending cabinet as leader of the House.

At staff level, a key appointment tells a similar story. In comes Dominic Cummings, the maverick, anarchic mastermind behind Vote Leave – hired despite being found to be in contempt of parliament for refusing to appear before MPs. Along with Cummings arrive several other veterans of the red bus campaign, a neat reminder of Johnson’s complicity in the £350m-a-week lie and countless other Vote Leave deceits.

In that same vein, listen closely to the speech the new PM gave in the burning afternoon sunshine. It made two key claims. First, the former leader of the 2016 leave campaign vowed that he would get Brexit done, “no ifs or buts”. After three years of what he described as lily-livered “indecision” and dithering, he would grab Brexit by the scruff of the neck and show that “this home of democracy” is capable of “honouring a basic democratic mandate”. He made it sound simple, as if all it needed was the smack of firm leadership and rhetorical firepower. There will be plenty of Brexiters, convinced that May only failed through her lack of conviction and inability to communicate, who will lap that up.

But note the second claim from the former mayor, as he promised that his government would be about so much more than Brexit, that he would get on with the pressing domestic issues that (by implication) his predecessor had neglected. Unlike her, was the tacit argument, he would not allow his administration to be devoured by Britain’s departure from the EU. And so he ticked off an agenda, doubtless tested and focus-grouped by Lynton Crosby to match the priorities of the voters Johnson will need, with policing, the NHS, social care and education at the top of the list (and in that order) – shaking the magic money tree as if piling up the deficit was no longer taboo. After that came a laundry list that ranged from free ports to blight-resistant crops to animal welfare to satellites.

The great intellectual colossus touted by his supporters had not managed to shape this list into a coherent argument (even though he’d had nearly two months to do it). Still the point was to show that his premiership would have a purpose beyond Brexit, and that the heart of the London mayor still beats within the Johnson breast. He still wants progressive voters to love him, or at least not to hate him – and so his speech referred to “equalities” and “LGBT” and made a promise to guarantee the rights of the 3.2 million EU citizens living in the UK.

This, then, is his pitch: that he can be both – both the Brexiter rabble-rouser of 2016, brutal in the pursuit of his goals, and the zipwire ringmaster of London 2012, provider of spectacle and amusement. The trouble, however, is that those two things will often collide. To take one example, Johnson said this sentence in his speech: “Let’s promote the welfare of animals that has always been so close to the hearts of the British people and yes, let’s start now on those free trade deals.” But what if one of those free trade deals involves the US, whose animal welfare standards so many British consumers find repellent? What then?

The larger point, of course, is that the project as he set it out rests on an impossible premise. For what was true of May remains true of him: Brussels will not easily offer a new deal, and parliament will not easily accept a no deal. Labour’s Keir Starmer is right to say that “no amount of rhetoric is going to get you through the reality”.

In which case, what we saw on day one of the Johnson era was the warm-up to an early election, with stubborn, vindictive Europeans cast as the villains and a new, hard-right, 100% Johnsonian cabinet, purged of dissent, lined up as the campaign team. Will it work? No one can yet know. But it’s worth remembering that Mayor Johnson and Vote Leave Johnson have one thing in common: when they faced the voters, they won.

•Jonathan Freedland is a Guardian columnist