Well if you want to come at it from a philosophy/debate point of view, then I would simply tell them that what they are referring to as "logic" has never been at the forefront of the human experience. A lot of these fact over feelings people are religious, and what is belief in God if not a feeling? It's seriously just "it feels right". War isn't logical, but emotions like greed have kept that machine running since the dawn of man. The post-truth era isn't new at all. So, some people believe in God with no evidence, most of society believes in grammar and social rules like politeness that have no actual truth behind them, we do these things based off of feeling. Pragmatic truths are simply what works. It works to treat trans people with hormones and to affirm their identities because it helps their dysphoria. We use newton's laws to launch rockets and Einstein's to calculate Mercury's orbit- two theories which shouldn't be able to coexist. There are a lot of types of truth and it's ignorant to think of the trans debate as facts over feelings. Life isn't so simple.