You claimed: "this only applies to television that is paid for by the license [...]"

That is incorrect. It doesn't matter whether the content was paid for by the licence or not. If you are watching live TV, you need a licence.

Please provide an example of a live television channel in the UK that has not received any funds from the licence fee. To my knowledge, even small local channels receive funding which is directly traceable to the licence fee, though of course the BBC gets the lion's share.