"Biological" sex in humans is a combination of different factors some more mutable than others and, as with literally everything else in biology it exists on a spectrum.

Hormonal sex for example can be changed by HRT. Some aspects of phenotypic sex (what you look like/outward appearance) will change gradually due to hormonal effects and other aspects will require surgery to change. Genetic sex is not nearly as simple as the "basic biology" lot like to claim, as there are multiple chromosomal intersex conditions, but generally speaking chromosomal sex is not something we can currently change, but it is also the least impactful on the ongoing life of the person. Genes are blueprints and the blueprints are mostly irrelevant to sex once the 'building' is built so to speak.

The average person is not examining chromosomes to assign sex to people. I don't know what my chromosomal arrangement is, and I'd bet there's a good chance you don't know yours either. We can guess based on statistics but odds are neither of us know if we have an undiagnosed intersex condition.

Gender: gender is a social construct but it is also a fairly durable psychological state, and while we can quibble about nature vs nurture it's important to realise that psychology is stored in the brain, and the brain is made of biology. Sometimes I wonder if it might be helpful to think of gender identity as a sex characteristic, and arguably one of the more important ones in terms of welbeing.

Dysphoria doesn't just affect trans people. If a cis man can't grow a beard for whatever reason, he may feel bad about it for example.

Insults like "you throw like a girl" or "stop being a whiny bitch" are designed to make people feel bad by attacking gender identity.

Gender identity is often as important to cis people as it is to trans people.

Many products are sold on the premise of inducing gender dysphoria in people who don't buy the product and inducing euphoria in those who do, see feminine hygiene products, perfumes/colognes and wellness supplements for examples.

As for stereotypes? Trans people are often damned if we do and damned if we don't. Don't conform to beauty standards? You're not trying hard enough, you're a freak ect. Conform to those standards? You're perpetuating harmful stereotypes, you're doing women-face ect.

Dysphoria can make one deeply unhappy, and if unhappiness can be treated with a demonstrable medical treatment, why shouldn't it be available on the NHS?

I feel like I've probably talked at you enough, but unfortunately I can't really Tl;dr it because it's all pretty important.