This is neither a sectarian literature nor an unpopular story. Ramayana has many versions. One of them is from Padma Purana. The story mentioned is present in Padma Purana Uttara Khanda chapter 242.
271-273. Then Rama, killer of his enemies, having gone to the Pampa lake killed there the demon Kabandha of a fierce form. The very vigorous one, having killed him, burnt him. (Then) he (i.e. Kabandha) went to heaven. Then the lotus-eyed Rama, having gone to Godavarl, asked her, "O Ganga, do you know my dear (wife) Sita?" That Ganga, covered with ignorance, did not tell him (anything).
274-275. Rama angrily cursed her: "Be of red water." Then being dejected she through fear, led by great sages, and being helpless and with the palms of her hands joined sought Rama's refuge. Then all great sages said to the eternal Rama:
The sages said:
- Ganga, purifying the three worlds, has risen from your
lotus-like feet. O lord of the world, you alone can free her from
the curse. Mahesvara said:
277a. Then the pious Rama, affectionate to those who seek
his shelter, said:
277b. May this Ganga, destroying sins, be free from my
curse merely by Shabari's bath and united with auspicious
278-279. Speaking like this, the very powerful Rama made, with the tip of his Sarnga bow, the Sabari-tirtha like Ganga and Gaya. In its water was the holy place of great devotees of Visnu. "Undoubtedly that form becomes venerable to the world."
So, Rama cursed Godavari to become of red water because she did not say where is his wife Sita. But then he immediately freed it by the request of Godavari and sages by creating a provision. Hence, Godavari is still a pure and sacred river. Its significance is said in many puranas. The video you watched was half quoting the story and left the second part about lifting the curse.