That's not the point they're making. The point is that they can offer definitions, and the left cannot. If you offered your summary of sex as a definition of "woman", you would be the left's enemy, because there is no allowable left wing definition more specific than "anyone who says they are".
Okay, let's indulge the façade of good faith..
I have no point of disagreement with the idea that biological sex is defined by the gametes a person produces. I would actually be more generous to conservatives and argue that there are in fact, multiple ways of measuring biological sex, including morphological sex.
The problem you seem to be having is understanding what "biological sex" in this sense means, or more specifically what it doesn't mean. Specifically, the societal role and position that comes with being a particular sex, for example of being a "woman" does not align in any way with biology. In fact, it predates the existence of biology.
Imagine a world in which an individual's sex was actually defined by the type of gametes they produced. Firstly, all children would be born with no sex. You'd have to perform surgical biopsies to find out what sex a child was. Even when those biopsies had been performed, some individuals lack the ability to produce gametes, so they would still have no sex. On the other hand, some people would be both sexes at the same time. Members of these groups might have entirely typical external anatomy, and the only way to tell them apart from more typical men and women would be invasive medical testing. It would be at least somewhat routine for people who morphologically resembled typical women to in fact be men, and a very small minority of people who resembled men would actually be women. In fact, it seems very unlikely anyone would bother finding out their sex at all unless they were trying to have children, since it would be completely arbitrary and irrelevant to anyone's day to day experience. It would certainly be irrelevant to the condition of being a woman, assuming such a condition even existed or was acknowledged.
Trans people in such a society would indeed have their sex clearly defined, but they would also be living in a society where not morphologically resembling your sex was sufficiently normal as to be largely unremarkable.
Again, the concept of being a woman predates biological science. It has never been possible to clearly define what being a woman means, the definition has always changed based on the prevailing beliefs of society at the time. Biology, unsurprisingly, has not provided any clear and simple definition of a woman. If anything, it has shown us how arbitrary our existing definitions of womanhood have always been, and how much more complex and variable nature actually is when compared against our crude, pre-scientific attempts to classify it.
What conservatives want is vindication of pre-existing and prescientific beliefs about sex. But in this society at least, being a woman is not and has never been a scientific concept. It means nothing more than being a person whom society at large says is a woman. Even the most radical forms of self-definition are merely asking for the right of an individual to have control over the arbitrary, mostly unscientific process that already happens.
