Radio is free and is paid out of general taxation

Not exactly true. In the UK it is either paid for by:
1) the "Radio Receiving Apparatus Licence" fee (aka TV licence). Radio used to qualify, and now exempted because of the difficulty of administering.
2) advertising
3) voluntary contributions and grants (community/hospital radio)
4) a mix of 1, 2 &/or 3 (even the BBC sell programs).

Drawing a parallel with the NHS. This is free (at the point of delivery - E&OE eg dentistry) but theoretically we pay for it via NI (National Insurance - a cluette there) contributions. Even the political hot potato, that is election giveaways, it is coming from NI, though which proportion visa vis dole & state pensions is a moot point.

Politicians like to tell us we are being given something when we have already paid for it. That is their conceit.

You will still need your own router to connect
That is the problem - crowded airspace. Solved poorly by carving up the allocated spectrum with channels and filling them with the packets. Eventually there is a traffic jam. Imagine - once everyone streams TV by default, multiple instances per household.

The thing is, once a resource like broadband is free, people use it without thinking of the profligacy and its repercussions. Like infrastructure (wot has to be financed - don't ya know?).