Investopedia seems to agree with you: "Neoclassical economics theories underlie modern-day economics, along with the tenets of Keynesian economics. Although the neoclassical approach is the most widely taught theory of economics, it has its detractors."
According to https://www.reddit.com/r/AskEconomics/comments/5wdup7/are_ec..., however: "The vast majority of all economists today work in the New neoclassical synthesis. This paradigm is essentially a combination of the best ideas from the New Keynesian and neoclassical trains of thought. The idea of separate schools isn't nearly as relevant as it used to be - these days ideas that work get added to the synthesis regardless of where they come from."
There's not much in so called "New Keynesianism" that Keynes would acknowledge as being consistent with his theories.
Be very wary of anything you read on /r/AskEconomics; contributors are regularly banned for asking challenging questions that point out deficiencies with the orthodoxy. It's essentially an echo chamber with the top level replies carefully vetted. What it highlights through that is the deep (and justified) insecurity that permeates the economic orthodoxy.
The Neoclassical school is laughably simplistic in its model of the world. To reduce an entire economy, with all its diversity and irrationality, to a handful of variables connected by simple relationships is frankly absurd [1], and really a reflection of economics' physics envy. They make strong claims about how the models are built robustly from micro-foundations whilst ignoring fundamentals such as irrational agents and missing variables, and blithely ignoring important problems such as the SMD result [2] which basically means only a single representative agent and a single representative commodity can be considered (I've seen models that claim much more than this, but practically the higher dimensions are immediately integrated out). That St. Louis fed model is considered advanced because they have 2 representative agents!
Moreover, the models don't maintain important invariants, such as stock-flow consistency, that absolutely must be true as a matter of accounting.
One might give some allowances for all these theoretical problems if the models validated and made reliable predictions, but when it comes to anything of importance, they're little better than a first order Markov model. It's frankly absurd, and a testament to the power of rhetoric and vested interests, that we've built so much of our political economics on such shaky foundations.
There's basically no alternative being seriously entertained in mainstream politics, even on the left, to what amounts to an academic justification for inequality, and that is why we are in the mess we are in, Trump and all.
Investopedia seems to agree with you: "Neoclassical economics theories underlie modern-day economics, along with the tenets of Keynesian economics. Although the neoclassical approach is the most widely taught theory of economics, it has its detractors."
This surprises me, as I thought Keynesian economics was the main view nowadays ("The central belief of Keynesian economics is that government intervention can stabilize the economy" https://www.investopedia.com/terms/k/keynesianeconomics.asp#...).
According to https://www.reddit.com/r/AskEconomics/comments/5wdup7/are_ec..., however: "The vast majority of all economists today work in the New neoclassical synthesis. This paradigm is essentially a combination of the best ideas from the New Keynesian and neoclassical trains of thought. The idea of separate schools isn't nearly as relevant as it used to be - these days ideas that work get added to the synthesis regardless of where they come from."