Keep the licence fee! Best value AV entertainment product I have ever purchased.
Think about American TV - the best shows are on HBO. Why? Because it is a pay per view / subscription channel not reliant on advertisers.
Shows like Sopranos, 6 Feet Under, Sex and the City would not get aired on mainstream networks because the content being strong (and part of the good quality scripts / dramas that they are), the advertisers would pull out around these shows. For example, Fox dropped Sopranos after the pilot, yet it grosses over $100 million a year. So the tame network dross on US TV is because it is controlled by advertisers.
Whilst British viewers might not be so reactive, there would still be a similar effect here. Also, as said above, nothing is free, we pay the price of advertising on the products in our shops - and I hardly ever watch ITV!
Want to dumb down British TV? Scrap the licence fee and have advertising on BBC.
Add to everything pointed out above concerning infrastructure, technology, news etc. We have the best broadcaster in the world with the BBC, most countries would love it - only us Brits fail to appreciate when we are well off!
As for the private health care analogy, well said! - yeah, stop funding the NHS and see how many nurses, doctors, physio's, radiographers, etc etc get trained to staff your privelidged hotels (the NHS trains them!); see how many multi-million £ MRI scanners you can find in private setups; and also, don't expect there to be a bed in your NHS intensive care unit when the private clinic co*ks it up, or an open A&E when you smash yourself up in your aspirational, I'm all right Jack (but sadly misguided) 3 series beemer.

Same issues - infrastructure, innovation, skills...
Just for info, NI contributions fund benefits and pensions, only recently was 1% earmarked for the NHS which is predominantly funded through general taxation. (I just say that because whenever NI goes up people bleat that they dont use the NHS so shouldn't pay, when in fact it is mostly about providing social security when their small businesses go t*ts up - to hell with the safety net eh?).
And And... and despite the problem of demand exceeding supply and the rise of consumerism; despite waiting 10 hours in A&E with your broken fingernail (because they are probably preventing someone else from dying); despite mistakes; and despite having some very poor "customer service" at times (as does your average organisation - like the many shopping horror stories on this forum) the NHS is still the envy of the world....like the BBC. (Please don't bother trying to tell me healthcare is better in Australia - it may be okay if you don't get chronically sick).
Whoops - was that off topic? Well, I think some of the issues are similar to the BBC thing.
Oh my god, I'm off on one now! Is there anyone left I forgot to offend