Still the phantom drive solution might not have been discovered because AI Jor-El on the artic ship wouldn't have known what kind of ship Zod was using. Jump to the wrong conclusions kind of thing.
I agree asking Jor-El would've been the smarter thing to do. However we can say it helped move the plot forward the way it happened in the film, but also Supes was curious about his people and probably didn't instantly think to treat them as suspicious threats.
A lot of things in the film are done for the sake of plot however I agree.