With that attitude nothing is independent: money has to come from somewhere. I think it's fair to say that every organisation has some amount of independence, some have money structures that allow them to have more than some others, and the BBC is an example of a situation where they can be more independent than an organisation that is more directly government funded.
The EU chip in a bit too. Not much as a % but it was £3 million of EU funds (i.e., European taxpayers who don't have a say in that either) between April 2011 and November 2013.
The biggest political influence is the board, not the budget. Since it takes a while to change the laws governing how BBC and similar agencies are run, I do not think the economics is the most interesting part.
The change from a Board of Governors to the BBC trust was a big one and not something done easily, so even there you have problems with affecting programming.
https://en.wikipedia.org/wiki/Board_of_Governors_of_the_BBC https://en.wikipedia.org/wiki/BBC_Trust
BBC news was not pro war in the run up to the Iraq war.
https://en.wikipedia.org/wiki/September_Dossier#The_45_minut...