Last month Mr. Miller’s group sued Arizona’s largest public school district, whose headquarters are in Mesa, for what the amended complaint described as “an astonishing situation that once would have been unthinkable.” It claimed that the district’s policy was to “encourage” students to change their gender identification and to forbid notifying parents if students chose to do so. The complaint then predicted several outcomes of such a policy, including sexual violence and genital mutilation.

The lawyer for the Mesa district, Robert D. Haws, declined to comment, citing the pending status of the litigation. But in his motion to dismiss the case filed this month, he said that the “parade of potential horribles” imagined by America First Legal was “completely off the rails.” Mr. Haws also said the school district was simply following a policy that forbids discrimination, including against transgender students.

Some of the litigation pursued by Mr. Miller’s group has suffered from questionable timing, at least from a legal perspective. In April 2021, America First Legal sued the San Antonio City Council for “canceling” Chick-fil-A (whose chairman, Dan T. Cathy, opposes same-sex marriage) by refusing to issue the chain a contract at the city’s airport. But the complaint was filed six months after Chick-fil-A, which has more than 40 restaurants in the San Antonio area, abandoned its plans for an airport location.

If the Chick-fil-A complaint seemed belated, a more recent filing by America First Legal might appear premature. In October, the group sued New York University on behalf of a first-year law student, a white male, who asserted that if he applied to the school’s law review he would most likely be turned down because of his race and gender. The school’s lawyers have filed a motion to dismiss, arguing that the lawsuit engages in “fanciful speculation” about a selection process that has yet to occur.

Kenji Yoshino, a law professor at N.Y.U., said that, America First Legal’s track record seems to suggest “more heat than light.” The group has yet to argue a case before the Supreme Court, and more than two years have passed since it succeeded in thwarting the Biden administration’s agenda.

Last week, during a federal court hearing in the Trump documents case, Judge Aileen M. Cannon rejected the argument made by Mr. Miller’s group in an amicus brief that the case should be dismissed because the National Archives and Records Administration lacked the authority to make a criminal referral. She called it “a red herring.”