Are Chinese taught in school how Chinese characters are formed (mainly phono-semantic compounds) or do they just bruteforce remember them? At least I know Japanese people don't learn phonetic parts of Chinese characters in school.
This is a textbook excerpt from the first grade, isn't it cute?
六书 is taught in 6th to 9th grade in mainland China, so I think the answer is yes. Good teachers often tell you how the character is constructed so that you remember it more easily.
As for phono-semantic characters, you don't really need a teacher to tell you about that. When you learn the characters 中 and 钟, it's obvious that they are related. Next time you see a character like 种, you would expect it's pronunciation is similar. It's just too intuitive. You observe the rule and apply them, with or without a teacher's help.
I think it's a bit like verb tenses in English. You observe the basic rules, you apply them to new words, then it fails because you meet an irregular verb, then you'll have to remember it. As time passes you observe that even irregular verbs have some regularities, then you try to apply your observations to new words, but they also fail at times. Chinese is like that, but messier...