This is a good idea. getmonero.org could have a page on the site where people can choose to mine for a while for a certain charity. It would be a really good marketing stunt

getmonero.org should be setup to do this.
It is an amazing idea.
We could choose also some marketing active charities and inform them about it. It would improve the XMR image.
Also donations to the development fund could be done on this way.
Whilst I appreciate that the concept behind that type of mining and see the potential in micro transactions through this, then it is an entirely bad idea to do any charity-related thing through official monero sites - such as getmonero. Why? Because Monero is apolitical. No matter which charity there will always be some sort of political underpinning.
What COULD be done is to do browser mining where all proceeds went to the monero foundation address instead - one could set it up when idle, and people could choose a cpu% or such. That would make it possible for people without technical know how or motivation to donate that way.
Unrelated to the official monero sites someone with the ability to carry through could setup donation sites - with the use of monero of course.
You do have a point but i think it is a bit splitting hairs here. What will happen imho if we do this is the following:
1) getmonero.org starts this and gets mentioned in many magazines
2) lots and lots of other sites copy the idea (of course one of the donations will be for themselves)
3) monero will be famous and then...
4) we decide to stop the mining 'experiment' because we want to be apolitical

You forgot:
1b) Potential debacle because the charity for purple kittens turns out to be a tax-evasion scheme, and the charity-donation for hungry african kids turns out to be going straight into the pocket of the tribe in power/government who repress the other tribes and use their poor hungry kids to fill their own pockets.
4b) ..and stopping it turned unsuccessful and furthermore energy was spent debating that "my charity is better than your charity". The skies are darkening, can you see the shitstorm coming?
Not splitting hairs at all. That's the kind of thinking that will get you in a ETH short squeeze. Name me one charity that would be a good idea to support officially with monero and I'll do my best to say why it's a bad idea.