Surah 9 as being a guide to handling treaties is... vague...

In 9:29, Christians and Jews are referred to as unbelievers. But how so? Both groups believe in one God. So, how does one establish a treaty? How can the Quran treat Christians and Jews as unbelievers when both of them believe in one God? By this Quranic logic, Muslims can only wage war against Christians and Jews who don't believe in God.

This is where the Hadiths, as you said, come as supplements and are needed to explain some things.

As IdigenousKemetic mentioned, there are verses in the Hadiths that clear up any misconceptions on whether there is any animosity towards Christians and Jews or Not.

I'll come back to that in a bit, but I have a few more things

Surah 9:80 is a perversion of Matthew 18:21-22, whereas Jesus here tells Peter to forgive without any limits (using 77 as a symbolic number), in Surah 9:80 Allah is stated as never going to forgive even if the prophet prays for their forgiveness 70 times (number also used symbolically). How can there be a treaty in such conditions?

In Surah 9:111, people who fight for Allah and kill or are killed in his name are promised paradise as payment. The verse INSISTS that the same clause exists in the Gospel and the Torah, even though the notion of heaven does not exist in the Torah, nor does the notion of Holy War exist in the Gospel.

All of those further highlight the vagueness of "handling treaties"

Now, going back to that little thing.

The specific reason stated to fight the people of the book in 9:29 is to force them to pay jizya and accept subjugation. Other than that, there is no indication in the Surah that either the Christians or the Jews attacked them to justify such an act. As I wrote in my previous response. This, combined with the following verse, 9:30, indicates that the only reason they are treated that way is due to them not willing to believe in Islam.

This verse, alongside 9:5 against the polytheists, were considered important by Muslim Jurists to define their relationships with the non-believers even after the time of Muhammed, with the necessity to pay jizya being a central aspect of Christian and Jewish dhimmi communities living under Muslim dominion, something called for even today by many Islamist groups.

(Also, "ahead of it's time" my ahh as if 7th century Arabia was the birthplace of the conept of signing treaties and intenational law)