The argument that being free gives access for poor people only works for those that live in London. You don't have to be very far away before the train travel gets so prohibitively expensive that if you are poor it is just not accessible. If you can afford the travel you can afford a modest entrance fee, even if you are saving for longer to have it.
If you use off peak fares you can only be there 11-3 ish, which with the queues is just not enough time to be a sensible trip.
The museums are national but the cost of travel means we are all paying for something only people in London/the city the museum is in can access at a reasonable cost.
We have a museum locally, smallish, costs £36 for one family, and that's no gift aid price. Many families can't afford this and a free museum in London is not much use when the train fare is so much.
Also, there has also been lots of people on the thread who say they don't go because of the crowds. The free entry removes access for these people.
If the museums are free they should be paid for by the local residents through their council tax, or have charges.