I agree with this assessment, and Mamdani's popularity in NYC provides some credence to this. Voters have wanted the Dems to move left since at least 2016, but the Democratic establishment routinely punishes those who aren't moving rightward.
Biden had a decent representation of left cabinet picks. But otherwise, the party has been pretty slow to change. Obama, in particular, gets remember as being progressive yet he truly was not. He took some antiwar stances and then failed to deliver on those promises. That was about the end of his left leaning policies.