The would-be Today complaint is classic 'complaints from both sides' territory.
Of course, Today is far from being pro-Brexit, and that's why this is such a classic instance of the fallacy of the 'complaints from both sides' argument.
Lord Adonis is talking nonsense about Today. The evidence runs in completely the opposite direction. His complaint, therefore, has no credibility because it has no substance behind it. The complaints from the likes of News-watch, on the other hand, have credibility because they have plenty of substance behind them.
Of course, he may have a point about Andrew Neil on Twitter (though not about his TV programmes). But if we start sacking BBC presenters because of their expressions of opinion on Twitter we'd lose a heck of a lot of BBC presenters. And if you add BBC reporters too then the BBC would be half-empty by the time the Andrew Adonises of this world had finally completed their purge.
Just think about it: No more Jon Sopel, Hugh Sykes, Katty Kay, Jeremy Bowen, Mishal Husain, John Simpson, Nick Byrant, Samira Ahmed, Nick Robinson, Paul Danahar, James O'Brien, John Sweeney, Kim Ghattas, etc, etc, etc.....
.....Hmm, come to think of it, maybe Lord Adonis has a point after all? Let the purge begin!