I live in a left wing country and local museums will gladly make people pay a lot to get scans of drawings or photographs, 150 years old, of their ancestors. They have no right to gate keep peoples personal history.
Since my comment above is my most-downvoted ever, I'll expand.
Somehow the building, collection and staff must be paid for. The opposite of your "no right" position is that there's no requirement for them to keep your personal history; they could throw it out. (Neither situation is absolute: the museum probably does have a legal requirement to maintain its archives, and there's probably not a legal requirement for access to those archives to be completely free.)
I used to work for a museum, and for each collection they knew the cost to maintain it. Every item in collection X costs £1.50/year to "keep", for example. This varies a lot, a collection of rock samples might be cheap, a famous painting in a humidity-controlled room expensive.
If the government provides funding, that influences how much revenue the museum must collect -- it could be between £0 and £AlmostEverything. In Britain, general access (i.e. having a look) is often free in the government-funded museums, as is academic/research access, but they try and make some money on the gift shop and other access, like yours.
Since Britain elected right-wing governments, the funding was cut significantly, with instructions to the museums to fund themselves by charging for services that were previously free.
> I used to work for a museum, and for each collection they knew the cost to maintain it. Every item in collection X costs £1.50/year to "keep", for example. This varies a lot, a collection of rock samples might be cheap, a famous painting in a humidity-controlled room expensive.
Is this information available to the general public? Do museums ask for donations to help with the upkeep of their collections? With government funding being slashed to such an extent, I'm not sure what's stopping them from doing that.
They ask for donations from wealthy donors, the general public, and appropriate research institutions (government or private). They employ professional fundraisers and so on.
Wealthy donors like to have their name on the new/refurbished building or a single project ("digitize these portraits"), but donate a lot of money for that purpose.
Research institutions will fund things necessary for the project, but it's difficult to get funding to make general improvements or maintenance. That's not their purpose. "Digitize the 19th century British portraits" sounds like a project, "migrate that data we digitized to CDs in 2000 to a new database system" generally doesn't.
The general public donate small amounts -- e.g. an extra £1 on their exhibition ticket, or £10 in the collection box when they leave, or an annual membership for paid-for exhibitions. This money can be spent fairly freely, so it's more likely to be spent on everything else -- maintaining buildings, rooms, curating collections, digitizing things that don't fall into a nice digitization project, and so on. In Britain, for the museums that have been free for a long time (including the British Museum), it can be difficult to increase these donations ("I already paid in taxes").
I can't see a by-collection breakdown in the annual financial report of the British Museum, or my former employer. Perhaps it would need an FOI request.