1- What is meant by not fully compiled? So was it just part of the Quran and not the full Quran or was it not organized the way we have it today?
2-Also, the companions didn't accept Imam Ali’s version because it had tafsir they didn't like. So they didn't take into account its organization. What about all the hadiths that say caliphs compiled it or that people were ordered to come with 2 wittnesses if they have heard a verse, are they rejected or do we accept them?

by

1- We do not know exactly how it was compiled. There are different reports, and many historical clues have been lost.

2- However, we do know that during the life of the Prophet (s), whenever verses would be revealed, they would be written on paper by some companions. Imam Ali (a) wrote all those verses on pieces of paper.

3- There is no strong evidence that all those pieces of paper were arranged into one full book. Remember that back then it was difficult to even make a book. So there were pages of the Qur’an not bound together into one book. Imam Ali (a) is the first who bound all the pages into one complete book. Yes, it seems that at the time of the Prophet (s) there was some sort of preliminary book because one hadith, for example, states that Jibra’eel told the Prophet (s) about Sura al-Baqarah verse 280:

ابن العباس: لما نزلت أخر آية على النبي (صل الله عليه وسلم) (وَاتَّقُوا يَوْماً تُرْجَعُونَ فِيهِ إِلَى اللَّهِ) فقال له جبريل عليه السلام (يا محمد، ضعها على رأس ثمانين ومائتي آية من سورة البقرة)

There must have been some sort of book then.

4- Remember that the current common recitation of the Qur’an goes back to Imam Ali (a), since the recitation of Hafs goes back to Imam Ali (a). So in the end, Imam Ali (a) did teach people from his copy and it became the most common recitation.

If you read Arabic, you may check this page for more details:

https://amp.annabaa.org/arabic/theholyquran/2748

And also this page:

https://www.almaaref.org/maarefdetails.php?id=1802&subcatid=99&cid=21&supcat=5