There's also the question of data. In order to track discrimination against a protected group (as defined in the Equality Act) you need to be able to define that group and collect accurate information.
The inclusion of male people in the protected group of female people distorts the data. This is already a problem to some extent, because transwomen with a GRC are considered "for all purposes" to be women, pursuant to the Gender Recognition Act, and, subject to the exceptions referred to by OPs above, that has included treating them as belonging to the protected group of "women". [NB there are good arguments they shouldn't be but that is what has happened]. However there are only about 6000 people with GRCs so one would hope the data remains just about reliable.
If there is self ID so people can get a GRC with less or no gatekeeping the numbers of male people who are included in the protected group of women also goes up, potentially to the point that the data is so unreliable that discrimination against women cannot be tracked.
For example, we currently have accurate data on the numbers of women on company boards. We can see that women are massively underrepresented. We can analyse why that might be (eg pregnancy, caring responsibilities, macho culture, etc). We can try to take steps to improve the situation.
If any male person can get a GRC then how do we know how many women are on boards? How can we discuss the "motherhood penalty" when some of the "women" can never be mothers? If a woman claims sex discrimination has occured because she was denied a position on the board due to her sex, the company can claim there's been no sex discrimination because there is a transwoman on the board so they have no problem with women on the board. Currently, correctly applying the law, a transwoman without a GRC would count as a man on the board, so would be no answer to the claim of discrimination. With self ID the ability of women to name discrimination is whittled away.
Unfortunately when TRAs say self ID wouldn't change things they have a point, but not for the reason they think. What has happened over the last few years is that data on sex is very often not captured. Equal opportunity monitoring forms frequently ask for gender or gender identity instead. At the extreme end of the scale such data is explicitly gathered on the basis of self identification rather than biological fact. So we have massive data gaps where no-one will be able to track sex discrimination.
Self ID will enshrine that into law. It's nonsense.