That sentence is natural for a native speaker to produce, and anyone would understand it, but it's actually not grammatically correct.
The clause "You don't really care which one" is quite common, and uses ellipsis to avoid repeating words earlier in the context.
Let's start with a typical use for it which is correct:
If someone asks you which shirt you want to wear, but you really don't care which one...
In this case, "which one" really means "which one you wear". The "you wear" part can be elided because it's already in the first part of the sentence.
It also works with verbs that are implied by the context:
If someone asks you to choose between two restaurants, but you really don't care which one...
Here, we can easily understand that "which one" means "which one you eat at".
But in the sentence you found in that video, we can't know what to put after "which one". Put another way, we don't know what actions are appropriate for the "option" objects, so there's nothing that can fill this blank:
If someone gives you two options to choose from but you really don't care which one _____________...
THAT SAID, this is only technically ungrammatical because native speakers say this and understand it without even noticing, including the teacher in this video, who speaks very well. So, kudos to you for catching it, but it's not even worth a comment on the video.