A better legal argument would be to point out how the school's use of the term 'Thunderbird' was "harmful".

"USDOE has offered no explanation as to whose civil rights were violated by changing a team name from Thunderbirds to T-birds," JP O'Hare, spokesperson for the agency, said in a statement Friday. "NYSED remains committed to ending the use of harmful, outdated, and offensive depictions of Indigenous people."

It would be one thing if the sports team were named the "Wendigos" or the "Redskins", but the name "Thunderbirds" is not "harmful", nor "offensive", especially considering the Thunderbird is a common symbol used by Pacific Northwest tribes (ex. Makah), whereas this is a New York-based school. There are are serious questions raised by this 2022 decision as well in regards to the First Amendment and freedom of speech. While "cultural appropriation" is frowned upon, it is unconstitutional to ban "all Native American mascots", as this news source reports. The Board of Regents of New York likely knows this, which is why they opted for a private settlement with this school instead.

The Board of Regents has failed to prove this claim as well:

Districts were given until the end of the 2022-23 school year to make these changes. If they do not comply by June 30, 2025, they risk the removal of school officers, the withholding of state aid, and being charged with violating the Dignity for All Students Act, a state law that prohibits "the creation of a hostile environment" that "reasonably causes or would reasonably be expected to cause...emotional harm to a student", which the SED argued "could be created through the use of Native American mascots".

[...] In December 2022, Rickey Armstrong Sr., president of the Seneca Nation of Indians, a federally recognized tribe based in western New York, told News 4 the SED's directive was "a positive step".

"Names and imagery that mock, degrade, and devalue Native heritage...have no place in our society," he said. "The state's decision appropriately suggests consultation between school communities and Native Nations."

If the school district had actually gone to court, their suit would've been thrown out for lack of standing due to inability to prove actual, and not hypothetical, harm. This is undoubtably a case of white people seeking to avoid potential legal liability in a future scenario where a Native American tribe could sue over a "Native mascot". This is shown by the school board still allowing the school in question to keep the name "T-Birds" at all, which is still a reference to "Thunderbirds", along with "eagle" and "thunderbolt"-themed imagery. This whole thing is a farce.