No, Adam, I think you're the one getting mixed up here.
You're close to getting it - and that Venn diagram illustrates the point nicely - but not quite getting it.
People who are opposed to gender identity theory because they are right wing/religious are not gender critical. Far from it. These are people who uphold gender norms, many of which are found in their religious texts. The kind of gender norms they choose to uphold are largely linked to biological sex in a fundamental way and focus on roles and behaviours, e.g. women are subordinate to men, a woman's place is in the home, a man's role is to provide for his family, women should bear children, women should stay at home and care for their children and when they're done with that they should care for their elderly relatives etc. There is a huge focus on women's value being in their reproductive potential, which is why younger women of childbearing age are valued as objects and post menopausal women become invisible. Given the focus on sexual reproduction, it is also not surprising that the same people who hold those beliefs often oppose homosexual relationships, access to contraception and abortion.
They also uphold other gender norms which do not have to be linked to biological sex, but are nonetheless rooted in them. For example, women frequently - whether they realise it or not - wear clothing and makeup designed to do the equivalent of a male peacock's feathers, i.e. attract a mate. Women's and even little girls' clothes are designed to accentuate hips and breasts, i.e. indicating health and fertility, and makeup is designed to convey the impression of both health, e.g. using foundation to make your skin smooth and glowy, and readiness for sex, e.g. lipstick to give the lips a more intense colour and eye makeup to create the illusion of pupil dilation. Conservatives usually prefer women to dress in this way, which is considered gender conforming.
Right wing religious conservatives are not gender critical, they are pro gender, and they are clearly not feminists.
People who are proponents of gender identity theory also uphold gender norms, only different ones. They struggle to uphold the norms based around biological sex because no matter how he behaves, a male person can never bear children. The role of a trans woman is clearly not to be a stay at home mother and housewife; indeed, a trans woman who has transitioned in middle age may well have been the main breadwinner whilst their actual female wife was a stay at home mother and housewife. So this person clearly has not been "living as a woman" in that sense.
But what other sense is there? They can't bear children and I see no evidence that trans women are "living as women" in the sense that they stay at home and do housework, or care for elderly relatives, whilst the person on whom they are financially dependent goes out to work. So what is left? I'm afraid it's really only push up bras and lipstick, as inconvenient as that truth is.
To complicate matters still further, when a male person who chooses to wear push up bras and lipstick, and believes that in doing so they are "living as a woman" and therefore entitled to use women's spaces, what happens when they meet with resistance from women who want their single sex spaces to remain single sex? The answer appears to be, in most cases, that they default to behaving in a very male way. Some are aggressive and violent, inviting women to suck their lady dicks or brandishing placards saying "punch a TERF". Others merely continue to use women's spaces in the full knowledge that women have not consented to this but there is nothing they can do to stop it. This is not women's behaviour; this is men's behaviour.
So we see that, unable to uphold gender norms which are inextricably linked with biological sex, male people who wish to show that they are "living as women" often present in an over-the-top, stereotypically feminine way in terms of their physical appearance, but their behaviour tends to remain undeniably male, especially when challenged.
There are so many overlaps between right wing religious conservatives and gender ideologues, the most significant being the fact that they all seem to believe female people should bend to the will of male people.
Gender critical feminists do not believe any of these things. We simply believe that biological sex is real and important. A female person cannot completely identify out of the constraints placed upon her as a result of having a female body, just as a male person cannot comprehend these constraints because he has never experienced them. We reject the stereotypes and norms as far as it is possible to do so, and where it is difficult to avoid performing them, e.g. when newly postpartum, we at least recognise them for what they are. We are also broadly accepting of gender non conformity and non heterosexual orientations.
The only thing gender critical feminists have in common with right wing religious conservatives is that we believe a woman is something you are or are not, based on your biological sex, not something you identify as based on your inner feelings.
And frankly I am getting a bit sick of having it pointed out to me that gender critical feminists have certain beliefs in common with the far right (I expect we all also believe that water is wet) by people who actually have a lot more in common with those people than we do, notably the fact that they clearly believe male people should enjoy higher status than female people.
So please stop banging on about how problematic it is that we have certain things in common with right wing religious conservatives, and go and make the same point to trans activists.
While you're at it, you could ask them whether they think it is problematic that both they and neo Nazis like to engage in the violent persecution of groups of people they think are beneath them.