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tri“Could the present perfect be called the most "important" verb tense?”
You could call it that if you wanted to, but I don't think that would accomplish anything. Why would you want to do that? Did your teacher ask you to write an essay on this topic? It sounds like that sort of thing. No matter which tense you choose, there might be good reasons to call it "the most important tense". Any tense you need to express yourself is "most important" at the moment you need it!

tri“Are there any advantages to teaching the present perfect in any particular order?”
In any particular order? You might want to be sure your students are comfortable with the verb
have first. Is that what you mean?
CJ