“I hate they/them. Use other pronouns.”
“Ok, you can use *proceeds to come up with the weirdest pronoun you can think of* ” and watch how fast they realize they/them isn’t that bad. You could also tell your family that they/them has been a singular pronoun since at least the 14th century so it's nothing new. Hamlet by William Shakespeare is an example you can give as an old source that uses they/them as a singular pronoun.
“Why are there so many labels?”
You’re right. Why do we have words for spoon, fork, and knife when we can just call them eating utensils? Why do we have words for bench, couch, and chair when we can just call them sitting spots? Why do we have words for shirts, pants, dresses, socks, and shoes when we can just call them clothing? Why do we have words for soda, juice, milk, and water when we can just call them drinks? People like to come up with a variety of words to explain specific objects and things we experience. Having quite a few terms to explain different sexualities and genders is no different than having words for can, bottle, cup, and bowl to explain different objects that can hold liquid.