Yeah, there is room to critique the methodology, but it appears basically sound. The full paper includes the full ideological spectrum; the centre-left/centre-right columns were only cut to make the message of the graphics clearer for the NYT.
While it would be nice to have more ideological detail than a simple Likert scale self-report, there’s no reason to think that the current methodology introduced any particular bias to undermine the conclusions. The picture is loosely focused, but should be undistorted.
The reason why the study went viral is because it accords with theory and history that is well-known to the left but ignored/denied by the centre.
Theory:
History:
…plus the other examples listed by Ben Norton or the NYT.
The liberal centre tends to be significantly more white and middle-class than the socialist left, and the white middle class is the demographic basis of fascism. Fascism is an ideology based in the suppression of the working class and the defence of fading privilege.
And looking at the graphs I saw what I expected - that center was the real outlier in most cases. It’s not that the further you get from center the more supportive of democracy you are, instead it’s that “center” a problem point, moderate left or right views aren’t implicated (though in the US “moderate” is a synonym for “centrist”).