Say the BBC made a great series in 1978 and paid the composer of the music for enough rights to air it in the UK, but not for media which were not yet invented at the time. The composer is now dead, the heirs moved to NZ and Italy, the will (written in 1982) doesn't make quite clear which heir inherited the internet rights, and none of the heirs really care to pay lawyers for the sake of a £500-1500 sale. How do you propose to arrange acquisition of the rights?

This isn't even very farfetched. A lot of old contracts were written without consideration of the streaming possibilities that eventually ensued. BTDT.

This is seen across a wide range of government departments: moving away from big central (often previously state) providers to smaller local private providers.

Is it in the public interest?

What is the benefit to the public of the BBC spending the money necessary to reclaim the rights to their old content? I can't think of anything. The shows that have any value (eg Dr Who) have already commericialized. Everything else ... why bother? The obvious point is that it'd be nice to watch some of the old shows again, but really, out of the 97 years of content across radio and television probably 99.9999% would never be looked ay by anyone.

Just as a parallel, there's millions of hours of old content on https://archive.org/ that never gets looked at. Why would the old BBC content be any different?