Good question
The Beeb own the copyright, so theoretically, you're thieving.
But you make a good point.
My take is that if you have a TV licence, I don't believe you'd be prosecuted.
Hardly good
PR, when as you say, we effectively own it.
If you haven't got a licence, you could expect to be in trouble.
One point I would contend is why you can't keep it forever.
I've never heard of anyone in Britain being done for recording Beeb material.
They're only interested in getting the licence fee.