Your beliefs surrounding the word neoliberalism continue to be bizarre. It is not an attempt to put center right and center left into the same box. It is a well-documented historical phenomenon. Hayek et al. called themselves neoliberals. The Atari Democrats also called themselves neo-liberals.
The shared framework is their support for market fundamentalism, deregulation, privatization, austerity, free trade, and globalization. It is just a fact that Clinton continued us down the route of neoliberalism started by Carter and advanced by Reagan.
1. I see no evidence that either Hayek or Atari Democrats called themselves neo-liberals. The term neoliberalism wasn't in circulation until the 1990s, except in leftist academic circles. This is a tiny point, as the labels mean very little.
2. Here's how I see the folks you call neoliberal:
***Not market fundamentalism; market realism. If Clinton was a market fundamentalist, he wouldn't have been so strong on antitrust (within what the courts allowed). Same with Obama. I think everyone acknowledges that markets, when they work properly, create prosperity that other forms of economic stewardship cannot. The difference between liberals and conservatives is that liberals are much more keen to detect market failures and advocate for policies to address them.
***Not deregulation but right-sized regulation: Reagan had a hostility to regulations. Clinton had a hostility to inefficient regulations. That's why Clinton retained the Reagan executive order on cost-benefit analysis. When I was younger, I thought cost-benefit analysis is bullshit. Then after law school, my view became something like: cost-benefit analysis is kind of bullshit, and over-reliance on it is bad, but there are no better substitutes.
Cap and trade, for instance, was a right wing idea. Liberals didn't embrace it because they were neoliberal. They embraced it because it fucking worked.
*** Austerity: that's just not part of the American story. Clinton's deficit reduction wasn't about austerity; it was about a peace dividend and higher taxes on the rich. I'm not going to comment on it outside this context as it quickly gets too complicated.
*** Free Trade: absolutely. Clinton and Reagan were free traders. Agreed.
*** Globalization: I reject this word because I think it is pejorative and doesn't have a well-defined meaning. If you're referring generally to the idea that nations should cooperate more and trade more, I mean sure but that's mostly overlapping with free trade.
3. Again, the problem here is emphasis and rhetoric. I've never said there are no continuities between Clinton and Bush, or Obama and Bush 43. But the differences are as important and your language doesn't allow for that. GOP are market fundamentalists. The Dems are not. They just aren't. There is no cogent meaning of that term that can describe any national politician associated with the Democratic party in my adult life -- at least not that I can remember. The GOP are deregulators. The Dems are not.
The problem with "neoliberal" as a label isn't that it is wholly inaccurate; it's that it effaces important distinctions, as I've described above. I mean, it's really weird to think of Clinton as a market fundamentalist when he:
A. Championed a "managed competition" health care system that would be government financed in most ways
B. Stepped up environmental regulations, to the point of even signing the Kyoto treaty.
C. Expanded the Community Reinvestment Act.
D. Championed and signed the FMLA
That's off the top of my head. That's the record of a market realist, of someone who believes markets can create prosperity in some conditions but will not achieve acceptable outcomes on their own without regulation.