Any 80-year-old institution based on the contours of a defunct 19th-century empire and largely held together by the charming drive of a 91-year-old woman is going to struggle to prove its modern relevance. Described once in the New Statesman by James Fenton as “one of the world’s least obnoxious institutions”, the Commonwealth can probably only ever aspire to faint praise. In an already overcrowded schedule of diplomatic summits, this is often seen as the “zombie summit”, a biennial gathering of whimsy that refuses to die.

Not surprisingly, the task of finding a thematic rationale for a Commonwealth summit of 53 nations, the first to be held in the UK since 1997, is not simple. The 70-strong Cabinet Office unit which planned the summit’s welcome chose the theme of our common future, and of welcome. Unfortunately, a sequence of decisions taken by the UK Border Force and former home secretary Theresa May on Commonwealth citizens up to five years earlier meant the headlines in the run-up to the summit were chiefly about rejection. The true motive of the British prime minister, it seemed, had been to a create a hostile environment for Commonwealth citizens, and to remove what they had assumed were unchallengeable rights. As PR disasters go, they rarely come much worse. Her Majesty was probably not amused.

But the May team has worked hard to get the summit’s agenda back on track by the time the heads of government convened for the first time on Thursday morning at Buckingham Palace.

Some ministers have long seen the summit as the moment to show that Britain can still shape the world after Brexit, partly by renewing old colonial friendships left untended during the years of European Union membership.

For many Eurosceptics, the discussion of a group of countries with Britain at its heart as an alternative to the EU has dominated thinking since 2010. Ukip presented itself as “the party of the Commonwealth” in the general election of that year, arguing for a Commonwealth free-trade area. In its 2015 manifesto it explicitly argued that the UK was part of a global community called the Anglosphere.

Such ideas also attracted Boris Johnson – now foreign secretary – who on a trip to Australia in 2013 said the UK had betrayed the Commonwealth when it joined the EEC in 1973, suggesting a free labour mobility zone with Australia.

Former colonial territories such as India, Singapore and Hong Kong could be embraced as part of an Anglosphere constellation, enabling Britain to exploit its past to strike new trade deals with the markets of the future, proponents argued. By the time of the EU referendum, these growing free-trade markets were regularly offered as a ready alternative to “declining” Europe.

This need to rejuvenate the Commonwealth as a trading bloc, a point referenced by the Queen in her opening address, led to an effort to persuade India’s prime minister, Narendra Modi, to end a decade of indifference to the Commonwealth and personally attend the summit.

Yet former foreign office ministers have told the government not to push the Brexit agenda too far. Many Commonwealth leaders do not welcome being part of an internal British dispute. Some, including the Canadian prime minister, Justin Trudeau, vocally opposed Britain’s departure from the EU. Smaller countries in the Commonwealth, meanwhile, fear losing a powerful advocate in the EU, knowing they now have to settle for Malta and Cyprus as their new Commonwealth champions in Brussels.

Few will also want to give the impression they will be soft touches in free-trade negotiations, or returning to an era of imperial trade favouring Britain. In a run-up session to the summit, the Indian deputy high commissioner, Dinesh Patnaik, said “the elephant in the room” was that Commonwealth countries such as India would only agree trade deals if they also included an agreement on the removal of the restrictions on the free flow of people, without which free trade would be likely to have little impact.

For most British industrialists, the EU remains the cornerstone for UK trade, representing 50% of Britain’s trade versus the 9% that goes to Commonwealth countries.

The summit is driven by four broad themes; a future that is fairer, more secure, more prosperous and shared. In many of the sessions – for young people, women and business – that preceded Thursday’s main summit, ideas, innovation, and projects were exchanged by often-inspiring speakers, all buoyed by the news of an occasional HMG grant or the visit of a glamorous royal.

Laudable announcements on cybercrime, malaria, plastics, free speech, ocean governance and 12 years’ education for girls added to the sense that this is at best a disparate development summit in search of a theme.

But hearing an authoritative David Cameron speak on Tuesday at Chatham House about how summits can be worthwhile if they are driven by a clear agenda, such as the fight against corruption, served to remind those listening, including some Labour frontbenchers, of one of this summit’s drawbacks.

Cameron, whatever his faults, takes issues of aid, trade and development seriously. By contrast, the current prime minister, unlike the royal family, has yet to show the same enthusiasm. With the Queen unlikely to attend another summit, since her overseas travel has ended, the onus on Prince Charles to keep the show on the road is immense.