Alice over at Wonderland has some interesting views on the subject -- and as a BBC employee, she's well placed to give her personal view in a lengthy and worthwhile piece. An interesting point ties in with both the BBC's Digital Britain and the recent findings that (unsurprisingly) lots of Britons are gamers:
Most importantly though, the BBC has an obligation to reach all of its license-fee paying households. If there are household members who prefer games to television and radio, then the BBC ought to be providing to them on the platforms and mediums they favour.
Perhaps we'll end up reliving history: in 1981 the BBC instigated the creation of a microcomputer to be used for educational purposes during the show The Computer Programme. This influenced the microcomputer revolution in the UK; if the BBC gets into gaming in the same way, perhaps the UK's attitude towards games could be similarly revolutionised. The BBC is in a position to turn game development and esports into a national pastime, if it so chooses -- the question is whether it ought to.