As you might expect from a cynical, right-wing operative who takes money from Charles Koch, this claim was a big fat lie. The board was very right-leaning then, and it's only become more right-leaning ever since.
Having followed the antics of the Intellectual Dark Web for several years now, I recognized at a glance that the board was packed with members of the Intellectual Dark Web, Quillette authors and other promoters of race pseudo-science, and a little research proved that all current members of the board but one have a connection to at least one right-wing cause or organization.
Then recently, the New Yorker published an article that made it clear that Rufo is probably the Kochtopus flunky of the year, thanks to his cynical and successful efforts to build up the previously obscure "critical race theory" into a Republican target of hatred and hysteria.
New York Magazine also ran a critique of Rufo:
Rufo’s role is clear. He takes critical-race theory as a concept, strips it of all meaning, and repurposes it as a catchall for white grievances. “The goal,” he tweeted, “is to have the public read something crazy in the newspaper and immediately think ‘critical race theory.’ We have decodified the term and will recodify it to annex the entire range of cultural constructions that are unpopular with Americans.” In an interview with the Post, he said the tweet described an “obvious” approach: “If you want to see public policy outcomes you have to run a public persuasion campaign.”
I think it's likely FAIR itself is funded by Koch, considering how many Koch employees are on the board, and also because the one issue that unites the FAIR board of advisors is opposition to critical race theory, as this table demonstrates.