The verse most people quote without reading
Bhagavad Gita 4.13 is fourteen Sanskrit words. Chatur-varnyam maya srishtam guna-karma-vibhagashah. "The four varnas were created by me according to the division of guna and karma." That is the whole claim. No mention of the family you were born into. No mention of your father's occupation or your mother's lineage. The two operative words are guna, the qualities that constitute your nature, and karma, the actions through which that nature expresses itself. Birth does not appear in the verse at all.