I fell in love with a picture at the Royal Academy summer exhibition. I’ve long been fascinated by the Tom Phillips project A Humument, which began in 1966 when he bought an obscure Victorian novel for threepence in a junk shop and began painting on the pages. The picture was a page from that, painted in blues, alongside text that speaks of windows and dreams. I gasped when I saw it. Although I cannot afford this piece of art (it’s £425), at least I got to go to a gallery and look at it, and that was a privilege.

Art belongs to us all. Not literally; more often than not, it belongs to some billionaire, but philosophically and spiritually. The Isleworth Mona Lisa, a painting of the same woman, Lisa Gherardini, as its namesake, is purportedly by Leonardo da Vinci. It is majority owned by an anonymous consortium, and spends most of its time in their Swiss bank vault. Recently, it was allowed out briefly for a public exhibition in Florence, and a custody battle has ensued – with a lawyer, Giovanni Battista Protti, arguing on behalf of its other part-owners that it should not be returned to the vault. “Whoever owns it has a responsibility towards humanity. It’s important it should travel the world, for the benefit of everyone,” he said.

We don’t even know who the owners are. That’s the art world: money talks, wealth whispers, and there are often fraught and complex histories of ownership to these art objects that in this case required the Panama Papers to untangle. Imagine if it turns out to be a Leonardo, as the owners claim (though I and many others have doubts). How greedy to hoard a piece of our shared history in a cold, dark safe, the opposite of the generosity artists show us in publicly revealing their souls.

Just because you love something, it doesn’t mean you have to own it. The privilege of looking should be enough.

Britain has an art export licensing system that some consider balanced in how it handles the competing interests of art institutions and art owners. We don’t have a French-style system where museums can pre-empt the sale of an object of national importance by matching the price, but an export bar can be placed on such an object. A temporary bar has just been placed on Turner’s watercolour The Dark Rigi, the Lake of Lucerne, valued at £10m. This, in theory, allows time for funds to be raised so the painting can be kept for the nation.

Export bar decisions are made by the reviewing committee on the export of works of art and objects of cultural interest. This is an independent body, serviced by the Arts Council, which advises the culture secretary on whether an artwork is of national, oustanding aesthetic, or academic importance, in which case an objection to export can be raised.

The system is far from perfect. The RCEWA’s annual report contains a breakdown of the treasures we have saved and those we have lost.

In 2015, a Cézanne masterpiece went abroad after the £13.5m asking price was not raised. Last year, Britain lost an album by the pioneering photographer Julia Cameron, and a rare Rubens portrait of an African man that was deemed to be of “outstanding aesthetic importance” and “of outstanding significance for the study of Rubens’ artistic practice”.

Two years ago, a Pontormo portrait was bought by the billionaire New York hedge fund manager Tom Hill. The National Gallery, along with ArtFund and other donors, raised the £30m asking price for the Portrait of a Young Man in a Red Cap, yet Hill turned them down, which led to calls for reform of the export licence system. The government launched a public consultation on a proposed legally binding mechanism to hold buyers such as Hill to their commitments.

Hopefully, this will be put into place. But it still doesn’t go far enough. The current system is designed to keep works in the country, not necessarily on public display. There is a 10-year period during which buyers who refuse matching offers are not allowed to reapply for an export licence; this should be increased to 25 years. And why not adopt the French system, where museums can pre-empt the sale? At the moment, owners are gaming the system by putting artworks beyond the affordability of museums. Institutions need more resources to work with donors and collectors and get to grips with the labyrinthine regulation system. And more transparency is desperately needed.

Someone told me recently that you don’t remember the feeling of standing in front of an artwork the same way you might remember where you were when a particular song was playing. But I can remember the dark, dusty church in Rome where I saw a particular Caravaggio – and feeling amazed by his light, vivid against dark corners, but also amused that you had to put a euro in the slot every few minutes to keep it illuminated. And at White Cube, smiling through tears at Tracey Emin in How it Feels, 1996. When asked if she feels she lacks the love that comes from having children, Emin says: “Well, you can love the whole fucking world, can’t you?” And I remember, at 18, feeling almost high in the Rothko room at Tate Modern; and actually high and rain-soaked and sad in front of Van Gogh’s final paintings in Amsterdam. I can’t buy any of the artworks I have loved. But just because you love something, it doesn’t mean you have to own it. The privilege of looking should be enough.

• Rhiannon Lucy Cosslett is a Guardian columnist