Sex discrimination and gender discrimination both exist and are different phenomena. We have to separate sex and gender in law and be clear when dealing with each concept which one is being looked at.
Employers may be discriminating on sex if they are avoiding employing females of childbearing age, or failing to promote them or give them development opportunities in case they go off and get pregnant. This discrimination would likely also be applied to transmen.
They may also and separately be guitly of discrimination on gender if people who don't perform the stereotypes of their biological sex "correctly", of either sex and whether or not they identify as trans, are treated less favourably. Eg by inistimg on skirts and heeled shoes for female employees.
I don't believe in the reality of gender myself but I strongly believe in the right to believe in gender and to be able to express that. So it's really important to be able to be very clear when providing facilities or opportunities which are legally and reasonably divided by sex, which is binary and immutable, that both the male and female options are explicitly and clearly gender-neutral - male people of any gender or none in one direction and female people of any gender or none in the other
The article's paragraph The framing of a gender critical belief as solely ‘sex is immutable’ conveniently omits gender. However, if the belief leads to manifestations such as ‘transwomen are men’, the belief elides sex and gender such that the actual belief seems to be ‘sex is immutable and gender does not exist/is not important/trans people are lying’. is totally misunderstanding (possibly deliberately) this point. Gender can be totally respected where appropriate, it's just that it's a totally different thing to sex.
What elides sex and gender is things like male people who have a feminine gender identity wanting to take part in female sport, when sport is legitimately and rationally divided by sex into male and female for physiological reasons which don't disappear with someone finding they have a trans identity. Therefore making sport more inclusive of gender diversity should be focused on ensuring that transwomen feel welcomed, included and not discriminated against or "othered" within the male category and that transmen feel welcomed, included and not discriminated against or "othered" within the female category.