Nine qualities define the brahmana's natural duty: inner calm, sense control, austerity, purity, forgiveness, straightforwardness, knowledge, experiential wisdom, and faith in the Divine. These are not inherited privileges — they are inner qualities that arise from a person's own nature.
Notice the order. It begins with 'shama' — the quieting of the mind — and ends with 'astikyam' — faith in a reality beyond the visible. Between these two bookends lie the practical disciplines: controlling the senses, enduring hardship, staying honest, pursuing both theoretical knowledge and the living experience of that knowledge.
Anyone in whom these qualities naturally arise is doing brahmana-karma, regardless of what family they were born into. The Gita's definition is entirely quality-based. It looks at what a person actually does and who they actually are — not at their last name.