Brahamsutra never talk about birth based caste, the section referred there has a different meaning for word “shudra”, and not the caste specific meaning. In the example of Satyakaama Jabala , he was judged by action of speaking truth as a “Brahmin”, even if caste was not known..that prove castes are not taken by birth
In the BrahamSutra 1.3.34- when the Poorapakshi ( Karma Kandi’s ) asked question that “Shudra are not eligible to study Vedanta”, there the Braham Sutra disapprove them by giving examples.
In this Sutra शुगस्यतदनादरश्रवणात्तदाद्रवणात्सूच्यतेहि।।1.3.34।। Raja Jansruti after hearing that Raikva was more learned & more respected from him. He ran towards to Raivka to learn that knowledge, but Raikva denied him as he wanted to learn that knowledge to get worldly respect not as true learner. ( based on the story in Chandogya Up 4.1.1 ) Raikva referred Raja Jansurti as Shudra, this word there means (शुचम् आद्रवति इति शूद्र: ) the “one with the grief” not the caste specific word.
Also it is mentioned in the same story, that when again Raja Janshruti visited Raikva with pure heart, Raikva accepted him as his student. Secondly if you see the next sutra ( क्षत्रियत्वगतेश्चोत्तरत्रचैत्ररथेनलिङ्गात्।।1.3.35।। ) it give us proof that Raja Janshruti was a Kshatriya by caste.
In the last sutra (तदभावनिर्धारणेचप्रवृत्तेः।।1.3.37।।) it is mentioned that Vedanta knowledge can be given to anyone, even whose caste is not known, even the sanskaras. Gautama accept Satyakama Jabala as his disciple.