I'm going to put two terms that are backed by the transgender community, at least the one's I've spoken with, below

Gender - A societal construct which expects certain behavior from each sex.
Sex - The biological difference between male and female.

Gender relies on the stable nature of sex, but it also relies on the unstable nature of prejudice and sexism. For example, what makes a society? A "society" can range from the world, a country, a city, a family, and even an individual. Meaning what one person considers as a gender for one sex, can differ greatly from what another person thinks a gender is.

Gender at best then, is a pre-judgement. It is an expectation that when one observes a biological sex, they will behave in a particular manner within a certain social group. At worst it is sexism, in which a person is inflexible in their expectations and demands that a sex act a particular way.

What then does it mean to be transgender? For clear and consistent vocabulary, it simply means someone who crosses the gender expectations of their sex within a certain societal setting. I would say that allowing people to be cross gender is in an ideal sense, appropriate. It reduces stereotypes and allows more freedom, which I usually favor. It can be entirely free of prejudice or sexism.

Does being transgender change your sex? No. Barring minor exceptions like extra chromosomes, sex is biologically defined most readily through genetics. Meaning no matter how one acts, dresses, or presents themselves, their biological sex is still defined by their genetic aspect. A social construct cannot change a biological fact.

Can a transgender person validly claim they are trans sex? No. If I claim that I act, dress, or look a particular way, and this necessarily makes me the opposite sex, this would be prejudicial at best, sexist at worst. Sex is not a cultural construct. A cultural construct does not have the validity to imply that they should be treated as if they are sex of the gender they are expressing.

So what does that mean for pronouns? Are pronouns identifiers of gender or sex? The traditional use of pronouns is to identify sex, not gender. "She wore a top hat, he wore a dress." The behavior or expression of the individual is irrelevant to the identification of their sex.

A requirement that someone be called a pronoun that does not fit their sex because of an expressed gender stereotype is wrong. It is a requirement that gender take precedence over biology. A subjective construct cannot legitimately challenge objective biology. To do so would be sexist, noting that a subjective idea of one's actions make you a particular sex. Can people choose to use pronouns in a gendered way instead of biological way? If the parties agree to it, that is fine. But there should be no expectation or demand that a person use pronouns in a gendered way when their use has been implicitly biological.

Should gender be the deciding factor in any decisions that are based on biology? Again, no. Sports for example is separate based on biological, not gender differences. Women's shelters or prisons are separate due to the fact that a man can physically force themselves on a woman and get them pregnant. Vice versa for women to men. There is nothing discriminatory in basing differences off of biology, and one's expressed gender does not change this.

I am open to hearing other opinions, but this is currently the only consistent way I can look at the vocabulary for it to make logical sense.

In sum:

Gender and transgender or social constructs, or subjective.
Sex is a fact, and objective.

Objective considerations trump subjective considerations. The desire for subjective considerations to take precedence over objective considerations results in prejudice or sexism.