That's true, they can cut bursaries and community work. But I'd expect that few schools would be spending 20% of fee income on that. Maybe the very old, wealthy ones, but they'd be using endowments rather than taking so much off fees.
For exactly the same reason that it would be a problem to drop fees by 20%: because that would be reducing the education they are providing the people actually paying for it (and who are also paying all that extra income tax) by a huge amount.
I certainly wouldn't be happy to have £3.4k of what I pay each year going on community work and bursaries, leaving only £13.6k being spent on my DD's education. Especially since 17k fees costs me an extra £13k income tax and NI, whereas £13.6 fees (if they charged for the education they were actually providing to DD) would only have cost me £9.5k extra income tax and NI.
That would seem pretty excessive: I work extra hours to earn £30k but only £13.6k of that actually going on educating my DD, £3.4k going to charity, and £13k going to the government?! I'd be pretty pissed off if the school was doing that!
(Then if the school stopped that charity spending to not charge the parents more as you suggest, the breakdown would change to £13.6k going on educating my DD and £16.4 going to the government . On top of the government not having to pay to educate DD. Pretty pissed off at that breakdown too)