Peace brother ...
'Teach' is a very broad term which, as you know, can mean to impart knowledge, to explain something and to basically give 'ilm'. Of course, the verses had to be taught by a Prophet, knowledge which was given to him by God (75:16-18). He made use of wisdom by upholding the Quranic verses and applying them in his present context with a view to address his specific societal requirements. This wisdom was only taught by God. But to say that these 'teachings' have been faithfully preserved in the works of men, centuries removed from source, mouth to mouth is a completely different matter.
Yes, you are absolutely correct that it is God that teaches. He taught the Prophet wisdom (hikmah) from the same Book that you have in your hands today. God can teach wisdom via that same Book today, if He so wills, to whomsoever He pleases.
In verse 3:79, God tells us what Prophets of God would have told their followers to teach. This is not a reference to extra books or anything written by man centuries later. The verse clearly answers the question and informs one that the only thing a Prophet will ask you to teach is scripture. Of course, wisdom can only be extracted from those verses. Many Hadith reports have nothing to do with the 'wisdom' from the Quran.
003:079
"It is not (possible) for any human being to whom God had given the Scripture and wisdom and the prophethood that he should afterwards have said to mankind: Be slaves of me instead of God; but (what he said was): Be worshippers of the Lord by virtue of your constant teaching (Arabic: tu'allimuna) of the Scripture and of your constant study of it"
With regards whether Wisdom is Sunnah, I hope you find the following article to answer your question
http://quransmessage.com/articles/hikmah%20FM3.htmI hope this helps.
Kind regards,
Joseph