sa sarvavid - he is the one who knows everything. It doesn’t mean he knows all the disciplines of knowledge and every single thing which happened. There are two types of all knowing, sarvavittva with reference to know every detail, and we usually refer this status to Īśvara. The other with reference to realities in general - sarvajñatva, knowing the reality of all that is here is one satya-vastu makes one a knower of everything. The second meaning is what the word sarvavid in this verse coveys.
The whole teaching is summed up as Lord Krsna praises this knowledge in the last verse.
iti guhyatamaṁ śāstram idamuktaṁ mayānagha | etad buddhvā buddhimān syāt kṛtakṛtyaśca bhārata ||15.20||
Bhārata (Arjuna)! The sinless one! This most profound teaching has thus been said by Me. Knowing this, a person becomes (wise), one who has buddhi; and who has accomplished all that has to be accomplished.
anagha - the sinless one, where certain amount of punya should have been accumulated in many lives to have the opportunity to learn śāstra.
This teaching told by Bhagavān is the most secret - guhyatamaṁ. Secret is something that we don’t know yet it becomes known after being told. But this secret about my limitless nature tend to remain secret for most of people (whose qualifications are still not enough) even after being told. By saying that Lord Krsna silently laying down the ground for the next chapter which conveys values to be cultivated in order to have a mind to know this secret. Another reason for ātmā-jñana to be called as the utmost secret because of the hiding place is in the seeker himself. The seeker who goes about searching is the very sought. This is why śruti has to stop us and say “you are the sought”, otherwise we will be forever seeking.
Whether we are searching within or outside, it remains hidden because in the very seeker is the sought. To know this fact that we require something from outside to tell us to stop the whole searching and make us see that we are the sought. It cannot come from the knower - pramātā himself, but only from an outside source pramāna - the very means of knowledge which capable of producing a jñāna-vrtti that gives rise to the destruction of the ignorance of you being the sought. It does all that without depending upon your will.
By knowing this, this person becomes buddhimān - etad buddhvā buddhimān syāt. Everybody has intellect, but the capacity to discriminate between satya and mithya is important here. This person also is called kṛtakṛtya - one for whom all that to be done is done. Jīva uses his will to gain all the qualification, go to the teacher and learn, doing śravana, manana, and nididhyāsana. By the end of process, the will has no purpose to serve anymore since one has owned up wholeness of oneself. Then prārabdha takes over totally and things happen as they should be. Only for a wise person, pure prārabdha is there. That is why in the fourth chapter verse 33 Lord Krsna said “all karmas, are resolved only in knowledge.”
Om tat sat