Is it a bug that Relative Dates are broken in Search and Smart Groups?
I think you may find that, although the search results show the relative dates converted to fixed dates, when you create a smart group from the results it will save the relative date range.  

That is certainly what happens in my live v4.4.

I have also checked this as best I can on a 4.7.10 site that has had no activity or changes on it since February this year. I have a smart group based on a search that used the relative date filter Previous month.  That group has no contacts now and if I go to Edit smart group search criteria... (via the Contacts link on the Manage Groups page) I see:


  No matches found for:
   ...AND Activity Date - greater than or equal to "01 July 2016 12:00
  AM" AND less than or equal to "31 July 2016 11:59 PM" ...AND.


but when I click on Edit smart group search criteria... for the second time it shows me the actual search parameters which include: 


  Activity date :Previous Calendar month