I don't know what video this is, but if anything he should first focus on the Beit Yosef. The Shulchan Aruch is a condensed version of the Beit Yosef written in a more simplified manner. So I can only ask you: did he even talk about the Beit Yosef?
I disagree that Orthodoxy was born then. One might argue that the foundations of Orthodoxy were laid there, but then one could argue about almost anything in Jewish history that they were to become the foundations of Orthodoxy. There are a couple of reasons for my disagreement:
1. While Rabbi Karo did send out his book all over, the Ashkenazi reaction was led by the Remah (Rabbi Moshe Isserlis). The Remah, we know, had been about to publish a similar book and at first had been disheartened to learn that his work and toil was for nothing because he had been beaten to the (imaginary) print deadline by Rabbi Karo. But then he noticed that Rabbi Karo hadn't used Ashkenazi sources. So he turned his book into a commentary on Rabbi Karo's writings where he noted the Ashkenazi customs and laws. In the other words, even if one would argue that halacha had just been about to be set in stone and universalized, a minute later the Remah canceled that out. Furthermore, most Temani communities continued following Rambam's writings.
2. To say that halacha could no longer be modified is quite contrary to what halacha is, which is a key Jewish tool for navigating through all aspects of life. Life changes, and so halacha must be reactionary and malleable. Rabbi Karo himself fled from Spain to Israel, so he knew a lot about change. I very much doubt he thought he could freeze halacha. That would necessitate freezing all of reality. What was relevant in the Roman period was not in the Byzantine period. What was relevant in the Late Mediaval was not relevant for the Early Rennaisance.
3. Finally, "Orthodox" - as opposed to what? The term Orthodox necessitates the existence of a prominent Un-Orthodox, or Heterodox. The currently accepted model for the birth of Orthodoxy is as an Ashkenazic reaction to the Haskalic-Reform (and Conservative) breakaways (and Ultra-Orthodoxy was given a major boost after and because of the Holocaust). The Karaites by the time of Rabbi Karo weren't so prominent as they had been in the Gaonic and Early Rishonic (=Early Mediaval) periods, and didn't pose much of a risk (except when they involved themselves in local politics). Various one-time messianic movements were by and large specks of dust within the large Jewish historic framework. Traditional Judaism was not on the brink of disintegration. Creating an orthodoxy would mean wrenching things back into place. Back from what?
As a side-note, there are many points to be made about Rabbi Karo offering both new approaches and bringing back old approaches to law. For example: A differentiation between Zohar and fully legal sources - new; a return to the majority rule of the Sanhedrin - very old; simplifying halachic texts - somewhat old (return to Rambam's approach), and I'm sure there are more (I'm not an expert, although I recently heard of a doctoral dissertation written on Rabbi Karo's approach to halacha, which sounds like something I would like to read some day).
