I did read somewhere that the racks and bookshelves get bugged if you visit the house before purchasing decorations, I did this at the Breeze Home and is the only one, so far with the weapon's rack not functioning. With the other two (Riften and Solitude) I just went there until I got all the decorations and seem to be working just fine. But I don't really know

This is actually right! I've tried in Whiterun and Riften, houses that I had both visited once to see how it looks like before purchasing any upgrade; none of the weapons rack work. Now I just fast traveled to Solitude, became thane by finishing Elisif's request, bought the house with all its upgrades; all weapons racks work correctly. However the mannequins there seem bugged, the clothes don't appear even after re-entering multiple times, but this should be fixed in 1.4 as well from what I've read. Too bad we can't place the strange amulet on them to get rid of it.
I then placed the staff on a rack (must be equipped on the right hand), took it back and my quest updated itself asking me to bring it to Enthir. Problem solved.
You try the rack over the bed or one in hallway? I didn't even notice one of em for the longest time...
I hadn't noticed the one over the entrance before but no it doesn't work either.
however, the weapons rack on the top of the main bedroom in white run should work, at least in my case.
Not for me I've taken the shield and the 2 swords from it but couldn't put anything on it after.
Thanks to everyone who answered.