The BBC, as mushroom’s quote shows, was effectively coerced into agreeing to take on the expense of the free licences - in principle - as a way of ensuring the licence fee was not frozen.
It’s not obliged to take it on, however: the government would have know full well at the time that the BBC was likely to refuse to take on the entire expense - after all it more then wipes out any licence fee increase - at which point there’d be a whole hoo-ha.
In short, the government got a bit of moral blackmail in while saving money, while the BBC kicked the can down the road and decided to take the flak.
Either way,I’m at a loss to understand the logic of the OP