"Keynesians" are not people that talk about the ideas Keynes had. It's quite common for most "ians" or "ists" to not push the ideas their name implies.

Yes, MMT is largely based on Keynes ideas, so they will say a lot of the same that Keynes actually said.

At the same time, "Keynesianism" is more of a political movement than a scientific school (although, it's a well informed one). That's why they can be right or wrong independently from MMT. "Monetarism" is also a political movement, that people often confuse or try to claim to be the same as MMT.