I think the key issue here is that the left and the right are speaking about two different things while using the same vocabulary to describe those things.
The left distinguishes between gender and biological sex. The right uses gender and sex synonymously, and makes the distinction with gender roles, stereotypes, and behavioral norms.
The left recognizes people with gender dysphoria as transgender, as well as anyone else who feels that they do not fit well within the gender norms (non-binary). The term "transgender" us used across the board.
The right recognizes people with gender dysphoria as transgender, but not those who feel that they do not fit within the gender norms. The term "transgender" is only used for the former because the latter is seen as a trend among people who are not comfortable with themselves.
My issue is that those who do not have gender dysphoria or a related disease who identify as non-binary are doing so only because of politics.
Gender dysphoria is a real issue that the non-binary crowd has, by and large, hijacked to fit their political agenda. I'm speaking here about people who simply do not fit into the gender norms of their assigned gender, so they utilize the current political climate to escape from that discomfort rather than actively attempting to change those gender norms (or ignore them because they don't actually matter). The right often calls these people "trans-trenders".
I understand both sides of this, but I do have a problem with people who have no medical problems demanding the world kowtow to them because of their insecurities.
If you have gender dysphoria, more power to you. I support you and hope you find what you're looking for.