meditrina I don't think they can redefine "the provision of education" as non-charitable.
No, I don't think that would be a good idea - as you say, it would cause chaos for everything from museums to literary societies.
The problem from independent schools is the shift from education, any and all education, being perceived as charitable, to education only being perceived as charitable when provided at a cost almost everyone can afford.
Rather than legislate on that, though, which would be difficult because of exactly the "well how affordable and by whom" issues that dog the bursary question, we should legislate to provide a route out of charitable status that doesn't require, as usual, that the assets be passed on to another similar charity.
Although I dare say arguments could be made for something broader, I think the sensible thing would be to limit it explicitly to independent schools - there are few enough of them and they're obvious enough that it would be practical to prevent abuse (wouldn't want it to be too easy for the secretary of the small literary society to pocket the £5k bank balance and then just happen to do nothing with it!)
You could even limit it to a one-time offer: "give up charitable status now, or forever hold your peace"...