That there is an epigenetic component is not what was contentious. My point being it is unknown which mechanism in the temperature-signal pathway is the root of TSD. KDM6B is proven to be a vital component in the epigenetic expression of DMRT1 but what triggers KDM6B? That is unexplained.

Also, I do admit "in humans, all babies start off development as females." is a bit of an exaggeration, but it is not incorrect. As seen in AIS individuals, as I highlighted in my post, without male differentiating hormones, human bodies, for the most part, develop phenotypically into female presenting bodies. So much so that prior to modern times and ready genotyping, the most common diagnoses of AIS occurred in mid to late puberty when the individual did not start menses or develop other secondary sex characteristics.

Many male organs are direct descendants/adaptations of female organs, the differentiation of which occurs only in the presence of male differentiation hormones.

So you asked why/how temperature-dependent sex determinism works and a lot of people mentioned greater whys and whats but not a lot about the the how.

And to be honest, we don't really know the exact microbiological pathways. There are three main theories https://www.sciencedirect.com/science/article/abs/pii/S0306456518304170 as to how this happens. https://www.ncbi.nlm.nih.gov/books/NBK9989/ But the main thing that scientists agree on is that there is an organ(s) that measures/monitors the environmental temperature, and then that sends signaling molecules that allow the gonads to develop as male or female, and then the gonads will produce additional sex determining hormones that will influence how the rest of the body will develop. One such base hormone (or a critical hormone in the signaling path) is aromatase. When scientists injected aromatase inhibitor into developing lizards, those lizards became male.

So the following theories are as such.

1. The yolk contains steroids (cholesterol based signaling hormones) that are released at certain temperatures.
2. The brain releases neurotransmitters as a general signaling hormone that are released at certain temperatures.
3. The gonads themselves react to the environment temperature epigenetically. This is the theory that is the biggest departure from the others as it doesn't specifically target a separate messenger hormone (even though one may or may not be present).

And while it's unknown specifically how these pathways form. The generic biomolecular pathway is pretty straightforward. Temperature can affect the shape of proteins directly. This is how some of your temperature receptors can figure out if things are cold or hot. A secondary method of temperature monitoring can happen by monitoring side effects of temperature. Chemical reactions vary based on temperature; generally reactions are faster the hotter it is. And by measuring the activity of certain chemical reactions (measuring the concentration of end/by-products, etc.) cells can determine the temperature. Then, that temperature protein can start a signal chain cascade to produce male hormones or female hormones depending on the environment.

And this is where the three theories take over. Scientists don't quite agree on where this temperature monitoring happens and which signaling cascades are involved.

But once those signals reach the gonads, they will influence the development into juvenile male or female gonads, which will then start to produce additional sex hormones that will further cement the development into male and female reptiles.

One thing that is important to note is that all fetuses start without sex differentiation. Embryos do not care about XY chromosomes, sex hormones, etc first. until a few weeks in. Male or female, all fetuses need lungs, heart, brains, intestines etc. This undifferentiated fetus stage is what allows fertilized reptile eggs to become either male or female based on these hormones. Fun fact, in humans, all babies start off development as females.

Hormone based sex determinism is actually very common and is one of many many many sex mechanism in even humans. One specific case is Androgen Insensitivity Syndrome. While it is a complex condition that affects many people, its base mechanism is that one of the sex determining hormones acts like the aromatase that I talked about in lizards. This hormone, androgen, is what allows human fetuses to halt development of female anatomy and promote male anatomy. Those affected by AIS are genetically male, but possess a lot of female anatomy and do not possess male anatomy because androgen could not act as a sex differentiating hormone. Due to mammalian XY genetics, however, their gonads are still proto-testicles and do not produce secondary sex hormones that promote the development of female anatomy. Nearly all people with AIS do not have uteruses or a fully formed vaginal cavity and are considered infertile. I hope I have not done the AIS community an injustice with my bullet overview of a complex topic.

The main point I want to get across is signal pathways, signal pathways, signal pathways. Complex multicellular organisms need a way to talk to itself and signal pathways is the method.

TLDR: Temperature -> temperature-sensitive protein -> signal hormone(s) -> sex differentiation -> additional signal hormones -> male or female.