The NHM is free to enter, some special exhibitions charge for entry and I think some require free booking to manage crowds. There is a very strong encouragement to make a donation though.
I really don't get what you find wrong about this.
My local museum started charging for entry a few years ago, along with a refurbishment, new exhibits, a bigger gift shop and a push to attract more tourists. So now it's horrible. I'm not sure what the unifying mistake is in both models, free entry and ticketed. I think the error might be in trying to serve the public.