The purpose of New Labour was to create a 'progressive coalition' that would shift the country to the Left, but 'left' in the metropolitan sense. This was called 'the Project' by insiders. Cameron's liberalised Tories can be seen as part of this process, while Gordon Brown's premiership locked-in fully the left-of-centre part of the Labour Party - i.e. the likes of Balls, Cooper, Benn Junior.
Blair and his backers - in my view intelligently - decided in 1994 to accept liberal globalism (including both thatcherism and extreme social liberalism - the two go together) and develop a social democratic strategy that would adapt the Labour Party to these new realities and cement a progressive centre ground that would be immovable. The Project succeeded and that is what we now have in British politics. 'Progressive' is of course a misnomer, an obscene abuse of a perfectly good English word, but it is used here to mean a sort of metropolitan leftism - a type of management that favours liberal capitalism while actualising certain traditional objectives of the Left.
I think most of the public are unintellectual and respond to politics in terms of mood music and the lies they are willing to believe, or at least give credence to. As the country tired of Blairism, it seems to me that Corbyn's approach to politics was a response, differing only in the smoke and mirrors deployed and the espoused methodology. Corbyn is portrayed as some sort of '1970s socialist', but Corbyn is part of the same metropolitan milieu as the left-liberals of New Labour. There are important differences, which should not be underestimated (especially on nuclear weapons, foreign policy, maybe education), but the rest is detail and overall a row between a Blairite and a Corbynite looks rather Pythonesque. Both agree in the basic project to destroy traditional Britain and create a "new, modern country", which in essence means a country that left-leaning urban dwellers are comfortable with, in which people like us are marginalised and at most tolerated and play no part in the mainstream, which remains 'progressive'.
But the public might not appreciate all this, and while I certainly agree that the Labour Party is in difficulty, Jeremy Corbyn has been put there for a reason. The right of the Labour Party may have put him there so as to neutralise Labour's Left, so that a more intelligent strategy can be resumed. The left of the Labour Party have put him there in the belief that he can appeal to the country and take the progressive project further, maybe in coalition with the SNP. We'll see who is right.