As others have mentioned, the museums did used to charge round about the late 90s. It was wonderful when it became free.
It used to cost about £20 for my daughter and I to get to London and that was usually all we spent when we went up for the day. I didn't drive then do it was actually easier to do stuff in London than in my own county.
We frequently visited the Science Museum, the NHM, the V&A and the Tate Modern. We could not have afforded to do it if there was an entrance charge. We took sandwiches, drinks etc - the museums all had places where you could eat your packed lunch so it wasn't just the lack of entrance fee that it made it feel like they weren't just trying to screw money out of us - unlike so many other family attractions.
We appreciated the hell out of it. My daughter is now studying art at uni. Maybe the frequent visits to the V&A, Tates and National Portrait gallery played some part in that.
She and her boyfriend went to the natural history museum the other weekend. She couldn't believe he'd never been. She loved the familiarity of revisiting the place. Seeing the giant golden globe and the 'earthquake room' were like seeing old friends.
So yes, you are being as unreasonable as it gets. But, to be fair, everyone's already told you that.