I liketo treat neoliberalism not as some kind of coherent political philosophy, but more as a set of interconnected ideas that have become commonplace in much of our discourse. That the private sector entrepreneur is the wealth creator, and the state typically just gets in their way. That what is good for business is good for the economy, even when it increases monopoly power or involves rent seeking. Interference in business or the market, by governments or unions, is always bad. And so on. As long as these ideas describe the dominant ideology, no one needs to call themselves neoliberal.
I do not think austerity could have happened on the scale that it did without this dominance of this neoliberal ethos. Mark Blyth has describedausterity as the biggest bait and switch in history. It took two forms. In one the financial crisis, caused by an under regulated financial sector lending too much, led to bank bailouts that increased public sector debt. This leads to an outcry about public debt, rather than the financial sector. In the other the financial crisis causes a deep recession which - as it always does - creates a large budget deficit. Spending like drunken sailors goes the cry, we must have austerity now.
In both cases the nature of what was going on was pretty obvious to anyone who bothered to find out the facts. That so few did so, which meant that the media largely went with the austerity narrative, can be partly explained by a neoliberal ethos. Having spent years seeing the big banks lauded as wealth creating titans, it was difficult for many to comprehend that their basic business model was fundamentally flawed and required a huge implicit state subsidy. On the other hand they found it much easier to imagine that past minor indiscretions by governments were the cause of a full blown debt crisis.
You might point out that austerity was popular, but then so was bashing bankers. We got austerity in spades, while bankers at worst got lightly tapped. You could say that the Eurozone crisis was pivotal, but this would be to ignore two key facts. The first is that austerity plans were already well laid on the political right in both the UK and US before that crisis. The second is that the Eurozone crisis went beyond Greece because the ECB failed to act as every central bank should: as a sovereign lender of last resort. It changed its mind two years later, but I do not think it is overly cynical to say that this delay was partly strategic. Furthermore the Greek crisis was made far worse than it should have been because politicians used bailouts to Greece as a cover to support their own fragile banks. Another form of bait and switch.
While in this sense austerity might have been a useful distraction from the problems with neoliberalism made clear by the financial crisis, I think a more important political motive was that it appeared to enable the more rapid accomplishment of a key neoliberal goal: shrinking the state. It is no coincidence that austerity typically involved cuts in spending rather than higher taxes: the imagined imperative to cut the deficit was used as a cover to cut government spending. I callit deficit deceit. In that sense too austerity goes naturally with neoliberalism.
All this suggests that neoliberalism made 2010 austerity more likely to happen, but I do not think you can go further and suggest that austerity was somehow bound to happen because it was necessary to the 'neoliberal project'. For a start, as I said at the beginning, I do not see neoliberalism in those functionalist terms. But more fundamentally, I can imagine governments of the right not going down the austerity path because they understood the damage it would do. Austerity is partly a problem created by ideology, but it also reflects incompetent governments that failed to listen to good economic advice.
An interesting question is whether the same applies to right wing governments in the UK and US that used immigration/race as a tactic for winning power. We now know for sure, with both Brexit and Trump, how destructive and dangerous that tactic can be. As even the neoliberal fantasists who voted Leave are finding out, Brexit is a major setback for neoliberalism. Not only is it directly bad for business, it involves (for both trade and migration) a large increase in bureaucratic interference in market processes. To the extent she wants to take us back to the 1950s, Theresa May's brand of conservatism may be very different from Margaret Thatcher's neoliberal philosophy.