I have a soft spot for specimen jars and skeletal remains. Museums of natural history, surgical pioneering or anthropological oddities have always struck me as equally suitable for lunch breaks and first dates as for serious study and research. As far as public and casually accessible encounters with mortality go, these kinds of museums are the most straightforward way of confronting the realities of human nature. But whether we should have this kind of casual access is now increasingly being questioned.

Telling history through displays of human remains presents a challenge for curators. They are responsible for contextualising exhibitions to ensure that the remains don’t become a dehumanised spectacle, while knowing they ultimately lack the ability to guarantee beyond doubt that their message will hit its target.

Last November, the Wellcome Trust decided context wasn’t enough. Its ‘Medicine Man’ collection, which had tried its best ‘to tell a global story of health and medicine’, closed after 15 years. Apparently, no effort to contextualise or to highlight marginalised stories of medicine could make up for the fact that a colonial legacy was granted the spotlight: ‘By exhibiting these items together – the very fact that they’ve ended up in one place – the story we told was that of a man with enormous wealth, power and privilege,’ Wellcome tweeted.

It sounded like a death knell for the private collections of long-dead white men. Meanwhile, the Royal College of Surgeons was busy with a six-year project redesigning the Hunterian Museum’s new exhibition space – which finally reopened last week – to tell the history of surgery and house the collections of the 18th-century anatomist, physician and surgeon John Hunter.

Born in Scotland in 1728, John Hunter became the founder of scientific surgery and operated on prominent figures of his time, including Adam Smith, David Hume, William Pitt, Thomas Gainsborough and even the infant Lord Byron. Hunter built a teaching museum in his house on Leicester Square, where he kept his 14,000 specimens gathered from all over the globe and brought in cadavers for dissection – possibly forming the model for the fictional Dr Jekyll’s laboratory and residence.

In 1813, the collection was moved to a newly erected building off Lincoln’s Inn Fields in Holborn where the Hunterian Museum has stood since. The museum’s central Long Gallery, designed to resemble the early modern Wunderkammer, is its crown jewel. Previously called the ‘Crystal Gallery’, the hall’s glass cases shine a delicate light on 2,000 physiological and pathological specimens, arranged in rows of pale yellow lumps of flesh, organ and bone. While I was surprised to spot jars labelled ‘Homo’ sitting beside a seagull stomach or a sloth foetus, this is illustrative of Hunter’s interest in comparative anatomy, grouping specimens together based on a shared level of complexity.

While the Hunterian Museum’s collection of human tissue enabled great scientific contributions, ‘it is also where some of those closely involved in the western “colonial project” developed sinister and awful ideas on racial theory,’ said Dawn Kemp, director of museums and special collections at the Royal College of Surgeons of England. A collection of human skulls from Hunter’s collection, for example, demonstrates the impact of geography and climate on humans’ physical variations, but proponents of scientific racism took these as proof that white people were biologically superior to other races.

Do museums still risk perpetuating these ideas? The Hunterian Museum holds its modern visitors in better faith, judging by the direct but low-key commentary of its placards on colonialism. Still, the ghosts of such misunderstandings certainly haunt today’s curators in concerns over how to minimise exoticism with context – and whether this is possible at all.

Oxford’s Pitt Rivers Museum, for example, has faced trouble with context. Its handwritten labels themselves read like artefacts – many of the 50,000 objects on display still bear their original labels from a time when words like ‘savage’ or ‘primitive’ were part of a common vocabulary. The museum is in the process of revising its public texts, but its decision to remove its tsantsas – ten ‘shrunken heads’ from Ecuador, some of which turned out to have belonged not to humans but to monkeys and sloths – cited audience research showing that, despite the written interpretation of the display, visitors mostly came away thinking that the point of the display was to show a ‘savage’, ‘primitive’ or ‘gruesome’ foreign culture.

While the purpose of displaying human remains is generally educational, the resemblance to the ‘cabinet of curiosities’ still stands. People enjoy gawking at oddities, which are often acquired by dubious means. The Hunterian Museum’s board of trustees decided to remove the skeleton of Charles Byrne, the 7ft 7in ‘Irish Giant,’ from display for the first time in 200 years. On his deathbed, Byrne requested to be buried at sea to prevent surgeons from getting hold of his remains for dissection. Hunter reportedly bribed a funeral-goer with £130 for his body.

Besides removing Byrne’s skeleton, the Hunterian Museum has added new galleries dedicated to modern medicine. Admittedly less visually stunning than the Long Gallery, the stories these rooms tell are more moving. Barbara Hepworth’s drawing ‘Concourse (2)’ captures, with great tenderness, a surgical team operating on her daughter in 1948, the year the NHS was founded. The final room displays the native heart of Jennifer Sutton, who underwent a life-saving heart transplant in her twenties. She was diagnosed with a congenital heart condition which her mother tragically died from during her own transplant operation.

These testimonies to human care sharply contrast with the story of Byrne’s stolen remains, which the museum still keeps for research, highlighting the disconcerting relationship between learning and looting – after all, the practice of anatomy in Hunter’s time was largely enabled by Resurrection Men who robbed the shallow graves of the poor. Now, two-and-a-half centuries later, there is still no clear guide for how museums should deal with controversial and questionably obtained objects. Earlier this month, the realisation that repatriated Benin bronzes might go into the private ownership of Nigeria’s royals forced governments and institutions to discuss whether removal and restitution are clear-cut moral obligations, or if they come with conditions.

But the Hunterian Museum concentrates on the life-affirming work of medicine rather than its doomed Eurocentrism. The care and cooperation captured in Hepworth’s drawing reminds visitors of the human compassion that undergirds surgery, and that there is often someone’s mother, father, child or spouse waiting anxiously for good results, which, thanks to pioneering surgeons and research institutions, we can increasingly hope to receive.