Straight out of the gate, Crash Course clarifies that they are not looking at the religious meaning behind the story, but its cultural influence. They also acknowledge that there are many different versions of the Ramayana, which is a good thing to note. The idea that the Ramayana is not being taken as a religious story, however, is the first red flag. What would you do if someone were to try to explain the Bible in such a manner? I doubt it would amount to much, unless we're talking anthropology. Them taking a truncated version of Valmiki's Ramayana from a textbook doesn't exactly help, either. They are interpreting an interpreter of a storyteller of a storytelling tradition.
The mother of Rama's brother doesn't convince the king of her son's better ruling potential, as Crash Course says. She just redeems a favor that he had promised her. The animation also glosses over the fact that Rama's father is distraught and heartbroken over Rama's banishment. The disfigurement is also glossed over, but I assume Crash Course is generally made for high school audiences, so I can see why they would remove that portion (despite showing a picture of it). It does not offer an analysis that takes caste, gender, etc. into account, instead giving a glossy version comparable to a truncated Odyssey that I was made to read in high school.
The application of the hero-story paradigm should also be investigated for a moment, I think. In looking at the playlist for Crash Course Mythology, I do not see an instance of Christianity, Judaism, or Islam being devoted a video. This might seem like 'inclusion', but to me seems to be treating Hinduism as mythology, which rubs me the wrong way in this instance.