No, Jesus would not have said "I am he". He clearly said "I have been" in the context of "I am". The context only carries so much weight when you're arguing for a grammatical basis for Jesus declaring to be God. Besides, Jesus would have to say "My name is I am" for the full effect of what you want to say. And with that said, as Prominent Trinitarians have noted, this traditional rendering is Modalist, not classical Trinitarian.
In other words, the meaning is clearly not the same. The context is defined by the grammar. The only reason one might believe Jesus declared to be God is because of the reaction to pick up stones, but this presumes that there's only one reason they'd have considered it blasphemous. The more likely reason would be because he was declaring to be a Divine, spiritual being who had existed since before Abraham. Besides, most people who focus and zero in on this verse usually completely ignore the actual context of what Jesus is trying to say from the last few verses.
Trinitarian Translators may be adding the "he" in John 9:9 to avoid the implications that would burst their precious proof-text that they've come to know and love. And they rarely ever talk about it either. In fact, I'd like to see a single Trinitarian site or scholar that has commented on John 9:9.
