Nobody is arguing that you should be required to use TWs and punished for not using them.

Nobody is arguing that you should be required to use TWs and punished for not using them. And no schools, functionally, have a policy remotely approaching that. And when TWs are used, they do not lead to the removal of material. That said, if I were teaching biology and a student asked me to TW evolution because of their beliefs, I would feel uncomfortable with that, too. Especially if I thought they were doing it because of their ideology, not because of their emotional or psychological needs. I completely understand how that scenario is bracing. My default is to presume competence and good faith when a student is requesting an accommodation. I would never ask a student to prove to me that a topic is traumatic. I don’t think I have the right to determine whole-cloth whether a trigger is legit or not. A lot of people have weird-sounding triggers out there. So, to answer your question, I would provide a TW for that student, because it’s a one-sentence warning with no value judgment. But I do not believe you should be forced to.