If anything can contribute to your misconceptions about jobs being tied to race it's the races DNA in a meta context. As in XI Galka had racial stats that of low mind and low intelligence but high strength. So they didn't make great mages but they did make really good warriors and monks. Meanwhile in the other extreme you had the Tarutrau who had high mind and intelligence and low strength. So they made for really good casters but poor warriors. This type of thing did carry over a little bit and sort of mattered until somewhere between Heavensward and Stormblood.

I sort of want to know what you're basing any of this on since just because a trainer is of a certain race doesn't mean that job is explicitly tied to that race. As you have Machi mages that are hyur and Elezen in the Lost City of Amdapor. Along with Hyur and Elezen Amdapori white mages. It's slightly hinted at that not every Nymian was a Lalafell. As almost every country that we know of isn't one made up of just a singular race.