>Temperature-based determination existed before genetic one.

Specifically, there is a hypothesis that the very first amniotes (reptiles, birds, and mammals are all amniotes) used temperature-dependent sex signalling, and only evolved genetic sex signalling later. There is some good support (phylogenetically) that the first amniotes did not have sex chromosomes but used temperature instead...and that sex chromosomes independently evolved multiple times within the amniotes.

The best model for why any species would adopt temperature-determined sex is probably the Charnov-Bull model. Simply put, the model predicts that in some species, the temperature at which they develop and hatch has a different effect on males and females, so temperature-dependent sex signalling gives you the best possible fitness in your males and females. For example, maybe a species lives in a place with cold winters, and lays eggs early in the spring. And let's say that species is best served by more females surviving the winter than males. If females develop in colder eggs, then eggs laid earlier in the springtime nesting season will all be female. That head start means the females will be bigger in the fall when the first frost hits. Males might then develop later in the nesting season and be smaller when the first frost hits, and a few more males may die during the winter, but the species as a whole preserves more females overwinter this way and improves its odds of surviving.

Of course, both can be true; Charnov-Bull could be the reason why the first amniotes used temperature as a sex selector, and part of the reason that some species have kept it to this day...but obviously many later evolved sex chromosomes (including us warm-blooded mammals).