Now for the correction: Klaatu didn't go to Professor Barnhardt's home with Patricia Neal's character, but with her son, her little boy. The boy was showing Klaatu around D.C. and that was one of the places they visited. The elevator scene you reference is between Klaatu and Patricia Neal's character, and yes, it is a very important scene in the movie.
Got it. I'm referring to a later scene.
SPOILER WARNING FOR ANYONE WHO HAS NOT SEEN THE MOVIE
The Day the Earth Stood Still
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Plot
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When Helen and her boyfriend Tom Stevens go out, Klaatu babysits Bobby. The boy takes Klaatu on a tour of the city, including a visit to his father's grave in Arlington National Cemetery; Klaatu learns that most of the deceased are soldiers killed in wars. They also visit the Lincoln Memorial.
Klaatu poses a question to Bobby, "Who's the greatest living person?" Bobby suggests Professor Barnhardt. Bobby takes Klaatu to Barnhardt's home, but the professor is out. Klaatu writes an equation on a blackboard to assist Barnhardt with a celestial mechanics problem; he leaves his contact information with the suspicious housekeeper.
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That's the scene you're talking about. But, later on ...
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Klaatu reveals his true identity to Helen and asks for her help.
Helen and Klaatu decide to visit Barnhardt. On the way, he tells her that should anything happen to him she must go to Gort and say, "Klaatu barada nikto." Their taxi is spotted and hemmed in. Klaatu is shot dead, and Helen rushes to the saucer.
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Those are the scenes where the taxi Klaatu and Patricia Neal are in takes this incomprehensible route to get to professor Barnhardt's house. They go every which way, and it doesn't make any sense.
Do you know that there really is a professor Barnhardt's house? Although most of the movie was shot on a back lot in California, there really is a house in DC that appears in the movie. It's north of Logan Circle. I've never been by it. I should make the effort to see it.
HTH. Thanks for writing.