Tropes are tools.
While we're at a point where certain people intentionally dig through TVTropes to find a trope they like and incorporate it into a character, the reason that tropes exist is because they're storytelling conventions with a long history in both verbal and visual communication. Zuko's haircut is actually relevant to the way his arc plays out because the changes to his character design are a visual shorthand to his character.
It's a fundamental part of good character design. Zuko's hair -- and clothing, and even posture -- changes throughout the show as a reflection of changes made to his character. You look at S1 Zuko and S3 Zuko and instantly be able to understand the broad strokes of his character development. Go back all the way to the early days of animation and look at the types of laws that Disney laid out to his animators to ensure that personality shines through; look at the way Renaissance artists would portray different Biblical figures and the kind of visual shorthand they would use to imply strength, righteousness, or vulnerability. Go back to the
Ancient Greeks and look at the visual shorthand they would use. To the Ancient Egyptians.
Go back all through human history and expression through art. They obviously didn't have 'character design appeal' as a metric back then, but humans have always had particular expressions of aesthetics and an inborn knowledge of what those aesthetics mean. While certain cultures have previously valued different things, or currently have their own cultural preferences, there's a surprisingly universal design language and meaning that can be easily codified as a 'trope'.
The intent behind talking about these kinds of tropes is to explore the intent behind the design and why certain decisions were made, which can in turn inform the intent of the character. It's all part of visual language, it's all important. People follow it for good, historic reasons and not because they read it on a tropes page and thought it sounded cool; people break these presumed laws for good reason, too. People didn't decide to start doing things Because Trope, identifiable tropes evolved from common or even universal storytelling methodology.
(Although Lily probably jumps like this because she seems to write around tropes rather than her writing just being a reflection of cultural tropes. There's a reason she puts together whole trope lists for her characters. She probably assumes that -- since it's how she works -- everybody else must just scan TVTropes and say "this sounds good" and just shoehorn it into their work.)
So... on the one hand, 'cultural appropriation' actually is a
really stupid conclusion to reach. While it's definite important in Asian cultures, in the West, you can trace this straight back to Samson and Delilah. I'm sure that if you dug through ancient myth in multiple cultures you would find themes with similar importance.
I was actually 100% with Lily on that response until she decided that FILTHY WEEBS were involved and then, like, what the fuck is wrong with you, Lily? Why did you feel the need to do that? Why are you
like this?