I'm a first year student, so I wasn't familiar with this structure, and I'm nowhere near going for HSK6...but I am familiar with another structure that felt about the same way to me: 到。。。去.
到 place 去 action is a similar two-verb two-object structure, it means "I am going to [place] to do [action], for example, I often go to a German conversation group at the local library, so if it's Wednesday evening and I'm heading out, I might say:
我到图书馆去用德语聊天儿。
I'm going to the library to chat in German.
This similar two-verb two-object structure also feels weird to me, and I think the reason it feels weird is that, in English, we would usually use an adverbial clause that starts with a preposition, as in "to chat in German", but in Chinese, we're combining the use of two different verbs of motion with different objects.
Now, I tend to think of the 用德语 in this sentence as being like a prepositional phrase meaning "in German", and it modifies the verb 聊天儿 (chat)...but notice that 用 is the verb "to use", so we're essentially using a verb to form an adverbial phrase. Maybe this means 用 is also a preposition, or maybe we use verbs to form adverbial phrases in Chinese, I'm not sure, but my feeling is it might be a bit of both.
Similarly, while I think it might not be the most correct way to think about it, I think it can help it feel more logical if you think of 到 as helping to form an adverbial phrase modifying 去 in this sentence.
Since 去 takes an action as its object (the thing we're going to do when we get where we're going), if we also want to say where we're going, we form the phrase "to the library" using the verb of motion 到, whose object is a destination.
You'll see similar things going on in sentences like 我坐地铁到图书馆 (I ride the subway to the library), where one could perhaps ask whether 坐地铁 is an adverbial phrase modifying the verb 到...it feels like perhaps instead 到图书馆 should be considered the adverbial phrase, at least to an English speaker, but it would be in the wrong word order then in Chinese, so there'd have to be an exception to the word order rules.
The bottom line, though, is that multiple verbs with multiple objects get used a lot in Chinese, at least when you're talking about going places.
My feeling seeing your example is that something similar is going on, that, like we use multiple verbs of motion when talking about going places in Chinese to add information about different types of objects (methods of travel with 坐/开/骑 etc., destinations with 到, reasons for going with 去), you can also sometimes use multiple verbs involving teaching and learning in sentences about education, with the different verbs taking different types of objects.
I think in your example, it's not confusing if you use the simpler form, but if the person being taught were some student with a weird name, this multiple verb form might make it much clearer where the line between the person's name and the subject being taught falls.
I expect which verbs can be paired up like this (or even tripled up like 坐。。。到。。。去 can be) is idiomatic and something you'll just have to get familiar with.
