Chait correctly points out that the left has historically been disappointed with the New Deal and Great Society, viewing them as lost opportunities. But he oversteps when he goes further to say that “neoliberal” is not only devoid of meaning, but that there was no essential shift in Democratic identity toward the end of the last century.
The difficulty of the term is that it’s used to described three overlapping but very distinct intellectual developments. In political circles, it’s most commonly used to refer to a successful attempt to move the Democratic Party to the center in the aftermath of conservative victories in the 1980s. Once can look to Bill Galston and Elaine Kamarck’s influential 1989 The Politics of Evasion, in which the authors argued that Democratic “programs must be shaped and defended within an inhospitable ideological climate, and they cannot by themselves remedy the electorate's broader antipathy to contemporary liberalism.”
Galston and Kamarck were calling for a New Deal liberalism that was updated to be made more palatable to a right-leaning public, after Reagan and the ascendancy of conservatism. You might also say that they were calling for “triangulation” between Reaganism and New Deal liberalism — or, at worst, abandoning the FDR-style approach.
In economic circles, however, “neoliberalism” is most identified with an elite response to the economic crises of the 1970s: stagflation, the energy crisis, the near bankruptcy of New York. The response to these crises was conservative in nature, pushing back against the economic management of the midcentury period. It is sometimes known as the “Washington Consensus,” a set of 10 policies that became the new economic common sense.
These policies included reduction of top marginal tax rates, the liberalization of trade, privatization of government services, and deregulation. These became the sensible things for generic people in Washington and other global headquarters to embrace and promote, and the policies were pushed on other countries via global institutions like the International Monetary Fund.