I have been speaking Greek my whole life albeit as a Greek American but I spent several years studying in Greece as well. I was always under the impression that I am looking for my wife would be translated as: " Ψάχνω για την γυναίκα μου. "
"Ψάχνω την γυναίκα μου. " I would translate as "I am searching my wife" although I got this marked correct because I figured out what you were looking for I still think the way you worded this sentence is awkward.
I have no quarrel with "τί ψάχνετε" that is of course what are you looking for?
The sentence used in this exercise though does not sound correct to me as translated. I've been speaking Greek my whole life and never heard this express that way. It might be a new usage of course since I'm old and I was a student in Athens in the 1970s.
After finding my dictionary, it seems to agree with you -- it has things such as ψάχνω ένα συρτάρι "search a drawer" and ψάχνω για έναν παλιό φίλο "look for an old friend".
One of its examples is [για] ποιον ψάχνεις; "who are you looking/searching for?" which may imply that για is sometimes used and sometimes not. But most of the examples with a concrete noun have για, which sounds odd to me but my language instinct for Greek must be incorrect in this respect. I shall try to adjust it :)
I learned Greek from 1995–1997; about 20 years ago, so after you but it's not completely fresh and up-to-date, either.
Thank you for prompting me to look :)