Here's a way that the government could be even less involved: don't DO that. Let people who want to show programs to a large audience find their own way to fund the production and dissemination of that material.
The courts, i.e. the gouvernment still has to enforce copyright in order for that to work. IOW, the government is always involved.
Say, by selling ads or attracting sponsors, etc.
Now you're under much more direct influence from the advertisers and sponsors. I'd argue strongly that under the current system the government has less influence over the BBC than advertisers and sponsors do on commercial channels.
Why should someone who doesn't want to fund a given program be forced to, under penalty of being dragged through court? I have zero interest in watching our many all-sports programming options (ESPN, etc).
Because we as a country thing that's the best trade-off. I've yet to see evidence that we're wrong. As a taxpayer, you have to fund all sorts of things you're not interested in. If you don't want to watch anything live, then you don't have to pay the live broadcast fee. You still get to enjoy the entire back catalogue via iPlayer, and in fact all the other channels offering on demant stuff.
You think the "best system we have" is for the government to be the enforcer in an arrangement where I'm forced to give them money anyway?
And (I'm guessing you're American), but American TV does not exactly convince me otherwise. Sure you have some great shows (much better on the whole), but your TV services are woefully uncritical of the government, unless it's along very strictly partisan lines. I have never, ever seen anything like this on American TV:
If you don't watch it, that's an interviewer on BBC news in the biggest time slot giving a very senior serving politican a very hard time indeed. The politician keeps weaselling out aswers and the interviewer simply does not accept. After being asked the same question again and again and again, the politician runs out of weasel words and finally admits what he did.
And this is not like the partisan screeching you get on Fox News. That is what a good news service ought to be doing.
The argument "tax is bad" is not enough to convince me of your arguments. Taxes are the price of civilisation, after all.