That wasn’t my interpretation at all, I don’t think she was proclaimed “unfit”. I wouldn’t even say it was about the Doctor’s discomfort or implied humiliation, it was simply a humorous take on first time TARDIS travelling, which felt very real.
I think part-time companions like this give the writing a greater freedom. There’s this expectation with the main companion that they have to be amazing, they’re expected to take to it pretty quickly, and possibly to save the day at the end. There’s obviously some leeway here - characters can have doubts, they can be a bit shaken, etc. - but there are limits here. After all, you wouldn’t want them to appear weak, would you? Throwing up would probably be out of the question (covering them in Starwhale sick is fine though).
But if you think about it, it’s a perfectly normal reaction. The Doctor is a piss-poor TARDIS driver and the ride there was probably rocky. And this is a life-changing experience, for anyone at any time, never mind their age. I’m not sure how I would react, but it’ll probably be closer to Courtney’s depiction than anybody elses.
It also adds another dimension to her character - we’ve seen her rebellious, sarcastic, curious bordering on nosey. She’s perceptive, fully aware of the “disruptive influence” she poses and she’s not afraid to challenge adults, whether they are teachers or caretakers, ranging from questions to downright insults. She’s got guts and she knows it. So that display of weakness was well-earned, in a way. It doesn’t make her less awesome, just more human.