I think the point is that Keynes often considered an economy in the liquidity trap (I also think he wasn't clear on this point and I'm not sure it was clear in his mind). By 1960 or so, economists had decided that a liquidity trap was a remote possibility. By 1980 it was views as Giffen goods are, theoretically possible but negligible.
I think you will notice (for example in this blog) attempts to define Keynesian economics not as economics in which employment may be below full employment because of low aggregate demand, but as macroeconomics with nominal rigidities. I think an accurate if clumsy definition would be macroeconomics with nominal rigidities or the possibility of a liquidity trap or both. Having typed that, I feel sure that the list of deviations of reality and therefore decent theory from the neoclassical model must number more than 2 (an Italian platitude is translated "there isn't twice without a third time").
Below find the usual rant which must bore anyone who regularly reads comments here. No I won't pollute the comment thread. It's over here.
On the other hand, I am not convinced by your newer Keynesian economics. I don't have suggestions for how to teach first year students, but I think there are statements in this post which don't correspond to reality.
“In the long run, monetary policy adjusts to achieve steady inflation, which means output goes to its ‘natural’ or Classical level. In the short run, monetary policy fails to achieve this, so we need to look at movements in aggregate demand to explain output.”There is not a Classical level of output. Output has a trend. It tends to grow. I make this trivial point to note that the idea that the trend level of output is not affected by aggregate demand shocks is entirely 100% an assumption made for convenience. In particular technology is not exogenous to the economy (people invent things, inventions don't fall out of the sky). Technological progress is not modeled, because it is very hard to model it. The result is that macroeconomists who focus on the cycle make very strong assumptions. "Exogenous to my model because I know I don't understand it" does not imply "unaffected by policy".
This works for any sensible monetary policy.
Also the short run can last decades. Continental Europe in the 80s had steady inflation and enormous unemployment. Actually right now inflation has been steady for years in all rich countries. However, output is far below the classical level. The claim that steady inflation corresponds to full employment or output at its classical level is a hypothesis. When Milton Friedman made his AEA presidential address (1968) it was consistent with available post World War II data. It is no longer consistent with available data. It is not a rough approximation to available data.
The competing salt water and fresh water schools have decided to agree about the long run. The shared assumptions are not supported by the data. They are much more important than the points of disagreement. But the agreed doctrine is based entirely on making assumptions about the long run in order to focus on the cycle.
Then the models are confronted with some sort of differenced or detrended data (say Hodrick Prescott filtered data). The strongest claims are about the long run, but long run relationships are not considered.