To some extent, but people from all sides have thought about this a lot in the UK, and other countries with a state television that is supposed to be as independent as possible. They are a lot more independent than most commercial outlets.

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