People do not produce exclusively sperms or eggs. Some produce both.
This is false. No humans produce BOTH. At the very most extreme edge of things, you might have a mosaic or chimera who has both testicular and ovarian tissue - but even if you overlook that this is a mixture of two individuals, they cannot simultaneously produce both sperm and ova. The level of testosterone required to produce sperm cells is toxic to ova. The level of estrogen required to maintain viable ova precludes the production of sperm.
You're not correct. Here's a Wiki, documenting the fact that there are true hermaphroditic individuals, at least some of whom are capable of reproducting.
https://en.wikipedia.org/wiki/True_hermaphroditism#:~:text=True hermaphroditism, sometimes referred to,containing both types of tissue.
1) they are not actually biological hermaphrodites, in the way the term is used by actual biologists.
2) they have both types of tissue present, but not both types of tissue functional at the same time
3) causes are predominantly chimerism and mosaicism, with a very few caused by DSDs - chimeras and mosaics contain cellular tissues from multiple sperm or multiple eggs - hence, technically not a single individual germ line
4) most people with ovotesticular disorder are phenotypically female, and only their ovarian tissue is functional
5) spermatogenesis has only been shown in TWO cases, ever
6) not a single on of the 11 documented cases of fertility has ever impregnated themselves
So let me reiterate what I said: No human produces BOTH EGGS AND SPERM.
Some species of animals do not have sex genes and temperature affects sex in some species.
No species of mammal has no sex genes.
No species of mammal has sex determined by temperature.
Humans are not reptiles. Nor are we clownfish. Nor are we algae.