[jambutty]

We buy a licence to view TV programmes by whatever means there are.
That licence money goes to the government.
The BBC is funded by the government from the proceeds of the licence fee. Whether the BBC gets the full amount or more or less is open to debate.

The Inland Revenue collects taxes and VAT on behalf of the government.
DVLA collects the Road Fund Licence on behalf of the government.
The manager of a newsagents, sells the goods, buys in stock and pays the staff wages on behalf of the shop owner.
The BBC collects the licence fee on behalf of the government. The BBC also has the authority to ensure that every household that has TV receiving apparatus installed in the house has a licence. If not they are empowered to taken the person to court. The BBC is very keen to get as much licence money as possible so that at the next round of funding negotiations they have a stronger case for increased funding.

Why is it so hard for people to understand that the TV licence fee has nothing to do with the BBC directly?

The BBC publishes that pie chart of how a typical licence fee is spent because it helps to establish that the BBC is independent of the government, which it is except for funding.

Even if the BBC goes down the advertising road we will still have to a pay a TV viewing licence fee. Read the back of the licence! It doesn’t matter which channel you watch you still have to buy a licence if you want to stay legit.