This is a bad sentence in English. Most people would read it as "I red it on the plane", not as intended "I reed it on the plane". In writing we would disambiguate it by saying "I have read it on the plane" and "I am reading it on the plane".
In general you are right, but context is everything. Here is an example where the sentence can be used:
Joe: "I hear you are reading War and Peace. It's such a long novel, when do you have time to read it ?"
Dagny: I read it on the plane to work each day (shorten to I read it on the plane).
As Groucho Mark said, "I once shot an elephant in my pyjamas, how it got in them I'lll never know."
Simple present standing for routine and repeated actions, I don't see why we should refuse it for this sentence while we accept it for thousands of other sentences.
By the way, "I am reading it on the plane" is an accepted translation.