"I believe that student."
Translation:나는 그 학생을 믿습니다.
It could sound rude to strangers because it can imply that the speaker has a higher position. We should not use 나 when we use 하십시오체.
it can imply that the speaker has a higher position
This is already well explained in the Tips and Notes section. I am aware the NIKL suggests us not to use 나 with 합쇼체, but this is just a suggestion for the sake of mutual respect and has nothing to do with grammar or our linguistic habits; the Standard Korean Language Dictionary itself has example sentences where 나 is used along with 합쇼체 or 해요체. Most of all, what do you think about this song?
"나는 행복합니다" is just a line of the lyrics that do not have to do with our "usual" use of Korean. (The lyrics of a lot of songs have errors that we usually do not point out and consider as "poetic license.") // Though the dictionary has such examples, you should think about what style "learners" need to learn first. In most formal situations, you cannot introduce or talk about yourself with 나. Do you think non-natives can figure out that odd use properly? I could not find such use in a lot of textbooks and lecture materials. Even so, sentences like this translation are used, which I won't deny. Nonetheless, at least Duolingo could add a noticeable warning to controversial translations. Now, I won't add anything further, since you won't change your mind.
It doesn't rhyme or anything so I don't see how it could be a "poetic license". Also I was just pointing out that it sounds perfectly natural to native ears. Learners most likely wouldn't have chances to mix 나 with 합쇼체/해요체 themselves, but we do frequently hear such use from rather older people. We usually learn a language for mutual communication, not unilateral delivery of information.
Do you think non-natives can figure out that odd use properly?
I wouldn't call it "odd", but as I said we did explain how it works in Tips and Notes, which you may call a warning.