One of the big takeaways was not to
over-value the knowledge I had gained learning English via immersion in an English-speaking culture, and conversely not to under-value that of the local teachers, who had gained their knowledge in the classroom. It's a cliche at this point to say that "street smarts > book smarts", at least in my culture. My teaching job taught me that there are situations where neither type of knowledge by itself is sufficient, and that both types have their place.
For example, as I mentioned I frequently ran into situations where I could tell whether a student's sentences were correct or not, but I struggled to explain why. One example from early in my teaching career was when students would place their adjectives out-of-order, for example "The German, red, old, large car..." instead of "The large, old, red, German car...". I intuitively knew that the former is incorrect and the latter is correct, but when students would ask me why, I struggled to articulate a reason.
But the local teachers on staff (i.e. native Chinese speakers) would chime in with "The order of adjectives in English is opinion, size, age, shape, color, origin, material, and purpose." They (the local teachers) still made mistakes in their English, but they had mostly memorized the rules from classroom study, and could recite them better than I could. Which was helpful to both us as native speakers (who wanted to give the students concrete answers to their questions) and to the students (who wanted rules to govern future scenarios they might encounter).
I was admittedly a bit cocky coming into that job, thinking I was qualified simply because I was a native speaker. I quickly learned that teaching a subject is a skill unto itself. It requires abilities like gauging levels of understanding by asking comprehension questions, and tailoring the subject matter to those comprehension levels, so as not to either talk down to the student or talk over their head.