[DTV]

The BBC managed for 80 or so years without such public input from staff members so I think it can last another 80 without it too.

Patrick Moore alone demonstrates this isn't true, publicly expressing a number of 'non-mainstream' political views over a BBC career that spanned seven decades. He's far from the only outspoken BBC presenter and, as you say, the BBC has managed just fine.

You can argue the difference between social media and newspapers, but Lineker's Twitter followership is about the same as the historic readership of some of the newspapers that presenters like Clarkson had a column in.