there are no structures in place to enforce ideological compliance with flat earth like there are with gender ideology
BUT THERE USED TO BE.
On April 12, 1633, chief inquisitor Father Vincenzo Maculani da Firenzuola, appointed by Pope Urban VIII, begins the inquisition of physicist and astronomer Galileo Galilei. Galileo was ordered to turn himself in to the Holy Office to begin trial for holding the belief that the Earth revolves around the sun, which was deemed heretical by the Catholic Church. Standard practice demanded that the accused be imprisoned and secluded during the trial.
The Catholic Church was so serious about matters of heresy that they created the Inquisition to prevent Christians from questioning the received truth of Holy Scripture.
On June 22, 1633, the Church handed down the following order: “We pronounce, judge, and declare, that you, the said Galileo… have rendered yourself vehemently suspected by this Holy Office of heresy...
Along with the order came the following penalty: “We order that by a public edict the book of Dialogues of Galileo Galilei be prohibited, and We condemn thee to the prison of this Holy Office during Our will and pleasure; and as a salutary penance We enjoin on thee that for the space of three years thou shalt recite once a week the Seven Penitential Psalms.”
Galileo agreed not to teach the heresy anymore and spent the rest of his life under house arrest. It took more than 300 years for the Church to admit that Galileo was right and to clear his name of heresy.
https://www.history.com/this-day-in-history/galileo-is-accused-of-heresy
Today we are free to believe in a Flat Earth or not. The only repercussion would be the damage to our reputation (which admittedly could have far-reaching effects). But don't pretend that it's always been this way, or that 
the belief that the sun rotates around the earth, not the other way round, was...not directly tied to any sinister goals
because the Inquisition knew that heliocentrism was a weapon of Satan intended to destroy Holy Mother Church.
TERFs are the Inquisition of today's transphobic movement, determined to root out the heresy of non-binary human gender. We need to oppose them with all our might.
After all, Inquisitors thought they were the good guys, too.