Yes, but Haidt is also wrong about a fair few things.
He undertook his research with the intent of finding conservative moral values (aka ‘values that bind communities together’ or words to that effect), which is fine if that was what he was going to then talk about, however, having found the bases of conservative morals, he looked at liberal morals through that conservative lens, compared the two and found liberal morals wanting. Which, no doubt, is a fair reflection of how dyed-in-the-wool conservatives see liberal morality.
He goes on to say that Liberals "only" use two of these moral foundations (Harm/Care and Fairness/Cheating).
However, it isn't true. What is true, from Haidt's own research, is that Liberals *prioritise* those two foundations ahead of the other three (Loyalty/Betrayal, Authority/Subversion and Sanctity/Degradation), but they do still use these. The difference is that conservatives use all five roughly equally.
Of course, in subsequent work, there is a sixth foundation, 'Liberty/Oppression', which is more recognised by Liberals than Conservatives, but is the foundation of primary importance to (American) libertarians (who, despite their lack of concern for Loyalty/Betrayal, Authority/Subversion and Sanctity/Degradation, still tend to band together with Conservatives).
I have a significant problem with what Haidt has done: effectively venerating conservative morality on the basis that it has more Foundations in his model. Of course it does, he explicitly set out to find the bases of conservative morality. This is begging the question. Indeed, his subsequent adding of Liberty/Oppression which is more a Liberal and Libertarian value (albeit defined in entirely different ways by those two groups) actually illustrates that Haidt overemphasized conservative values if his goal was to model all human moralities.
I think most people would agree that Harm/Care and Fairness/Cheating are the key elements of any morality, left or right. What conservatism does is apply a bunch of rules based on Loyalty/Betrayal, Authority/Subversion, Sanctity/Degradation as a means (one would hope) to achieve Fairness and Care and move away from Harm and Cheating.
If we agree that Fairness is important to any morality, then how we define fairness is important. This list of possible definitions comes from Lakoff (2002):
1) Equality of Distribution (e.g. one child, one cookie)
2) Equality of Opportunity (e.g. one person, one raffle ticket)
3) Procedural Distribution (e.g. playing by the rules determines what you get)
4) Rights-Based Fairness (e.g. you get what you have a right to)
5) Need-based Fairness (e.g. the more you need, the more you have a right to)
6) Scalar Distribution (e.g. the more you work, the more you get)
7) Contractual Distribution (e.g. you get what you agree to)
8) Equal Distribution of Responsibility (e.g. we share the burden equally)
9) Scalar Distribution of Responsibility (e.g. the greater your abilities, the greater your responsibilities)
10) Equal Distribution of Power (e.g. one person, one vote)
As a counterpoint to Haidt's research, I have found that Liberals have many more ways of defining Fairness than conservatives do. I grouped my participants together based on their self-stated political and religious affiliations and found that the left agree that 8 of these types of fairness were important, the right agree on 4. Secular individuals agree that 9 of these types of fairness are important, the religious agree on 4 (though, interestingly, explicitly monotheistic religious people agreed on 6). In other words, to put it back into a liberal and conservative frame. Liberals believe that between 8 and 9 of the above definitions of fairness are important whereas conservatives believe that between 4 and 6 are. Of note, only one of those definitions was held to be important by all political and religious sub-groups: “Equal distribution of responsibility.”
I think it fair to say that Liberals are using nuances within the concept of Fairness where conservatives are attempting to achieve Fairness via Loyalty, Authority and Sanctity. The problem is that Loyalty, Authority and Sanctity differ, depending upon who your authorities are.
On this last point, I’d highly recommend reading Robert Altemeyer's 'The Authoritarians' which illustrates forcefully the issues of Loyalty and Authority as moral values... and effectively predicted Trumpism a decade before he came to power: https://theauthoritarians.org/
I will read your linked article with interest.