It's important to note that the video in question is a presentation of the Christ myth theory, which is a fringe academic theory. Very few scholars give it any credence; in fact, New Testament scholar Larry Hurtado places Christ myth theorists in the same category as flat earthers and moon landing deniers. The theory's flawed methodology leads the narrator in the video to make the ludicrous claim, contrary to clear evidence, that Paul "never quotes anything that Jesus is supposed to have said."
But even among less extreme skeptics of Christianity, there is often a claim that, because Jesus' biography was not written down quickly and was embellished in the meantime, we know very little of the truth. Since Paul's letters are dated significantly earlier than the four canonical gospels and don't give much of a biography of Jesus, such skeptics frequently try to enlist his letters to demonstrate that even the earliest Christians knew almost nothing of the life of Christ.
In the following, I attempt to give a tour of Paul's letters to demonstrate his familiarity with the fleshly, earthly Jesus in a (hopefully) skeptic-friendly way; scholars debate whether Paul wrote all the letters attributed to him and many scholars don't believe that Acts quotes Paul accurately, so I've italicized references to such passages where Paul's authorship is disputed, and bracketed claims where there is no citation from the undisputed writings.
Paul's biography of Jesus, reconstructed
Jesus is both divine (Rom 1:4; Rom 9:5; 1 Cor 8:6, 10:4; Phil 2:6; Col 1:15-20; 1 Tim 3:16; Tit 2:13) and a Jewish (Rom 9:5; Gal 3:16, 4:4) descendant of David (Rom 1:3, 15:12; 2 Tim 2:8; Acts 13:22-23). [His ministry was foreshadowed by that of John the Baptist. (Acts 13:24-25)] He had many apostles, who were miracle-workers (2 Cor 12:12), including his brother James, John, Peter, and the twelve (1 Cor 9:5, 15:5-8; Gal 1:17-19).
He lived sinlessly (2 Cor 5:21), and then after his last meal (1 Cor 11:23-26) certain Jews caused his death (1 Thess 2:14-15; Acts 13:27). Following a Roman trial (1 Tim 6:13; Acts 13:27-28), he died by crucifixion (1 Cor 1:23, 2:2; 2 Cor 13:4; Gal 3:1; Phil 2:8; Col 2:14; Eph 2:16), in accordance with God's will (Rom 5:6-8; Gal 1:4) and with prophecy (1 Cor 15:3; Acts 13:29), as a sacrifice (Rom 3:25, 8:3; 1 Cor 5:7).
After his death, he was buried and then raised (Phil 3:10) by God (Rom 4:24, 10:9; 2 Cor 4:14, 13:4; Eph 1:20; 2 Tim 2:8) on the third day (1 Cor 15:4,20), and appeared to the twelve and to hundreds more (1 Cor 15:5-8; Acts 13:30-31). He is now at the right hand of God (Rom 8:34; Phil 2:9; Col 3:1; Eph 1:20; 1 Tim 3:16), as the Messiah and Lord (too many verses to count).
While I was preparing the reconstruction above, I consulted the work others who have undertaken similar projects, including Bart Ehrman (from his book Did Jesus Exist?), Mark Goodacre (Did Paul Think That Jesus Was the Pre-existent Son of God? and What Did Paul Know About Jesus?), Craig Evans (Jesus Tradition in Paul's Letters), F.F. Bruce (Paul and the Historical Jesus), and Bob Seidensticker (What Did Paul Know About Jesus? Not Much). Seidensticker is a blogger and a Christ myth theorist, Bruce and Evans are evangelical scholars, and Ehrman and Goodacre are New Testament scholars in the academic mainstream. You'll notice, if you follow the links, that my preceding summary is substantially similar to each of theirs. Unsurprisingly, Seidensticker is the most significant exception; his conclusion that "Paul doesn't even place Jesus within history" can be easily rejected, and his conclusion that "the Jesus of Paul isn't the Jesus of the gospels" shouldn't be accepted without a lot more scrutiny. Instead, Bruce's conclusion does a better job of cohering with the evidence:
What Paul has to say of the life and teaching of the historical Jesus agrees, so far as it goes, with the outline preserved elsewhere in the New Testament and particularly in the four Gospels.
Nevertheless, I must also agree with Ehrman (see more here):
Imagine what we wouldn't know about Jesus if these letters were our only sources of information. We hear nothing here of the details of Jesus' birth or parents or early life, nothing of his baptism or temptation in the wilderness, nothing of his teaching about the coming Kingdom of God; we have no indication that he ever told a parable, that he ever healed anyone, cast out a demon, or raised the dead; we learn nothing of his transfiguration or triumphal entry, nothing of his cleansing of the Temple, nothing of his interrogation by the Sanhedrin or trial before Pilate, nothing of his being rejected in favor of Barabbas, of his being mocked, of his being flogged, etc. etc. etc. The historian who wants to know about the traditions concerning [the life of] Jesus - or indeed, about the historical Jesus himself - will not be much helped by the surviving letters of Paul.
However, it seems reasonable to imagine that if Paul thought his letters would be the final word on the life of Jesus, he would have taken greater care to describe it in more detail. And for billions of Christians, his writings represent only about a third of the corpus of the New Testament, while the Gospels represent about half. So Christians have never relied exclusively on Paul for such details.