Traditional Vedanta commentators and Acharyas like Madhvacharya, Ramanujacharya, etc always held that Varna system is based on birth. However, Vajrasuchika Upanishad goes to another extreme and claims that neither individual soul(Jiva), nor body, nor birth, nor his knowledge, nor his deeds, nor his rites(Samskaras) can make a person Brahmana. The Upanishad also gives reasons why the above mentioned things cannot make a person true Brahmin. In the last verse, the Upanishad says, one who realized the Atman is indeed a real Brahmana. Such a statement is not supported by traditional Sampradayas of Hinduism. How do scholars from these Sampradayas view this Upanishad statement?
Assuming the Upanishad is authentic, the last line, as you mentioned, basically says, "one who realized the Atman is indeed a real Brahmana."
This Upanishad is explaining the etymological meaning of the word "Brahmana," which means "one who knows Brahman" according to Sanskrit grammar; that is the literal meaning of the word "Brahmana."
one who has divine knowledge
So, a Shudra, like Vidura, who has self-realization can be termed Brahmana insofar has he has qualities of Brahmanatva, but this doesn't make him an actual Brahmana by caste. What this means is that he can't marry within that caste, hear the Vedas, recite mantras, perform Yajnas, teach the Vedas, get initiated into Upanayanam, and other samskaras that are only for Dvijas by birth.
The whole point of initiation into Upanayanam, otherwise known as Brahma Upadesham is to realize Brahman.
The Rahasya Traya Sara of Sri Vaishnava acharya Vedanta Desika discusses this: