Reinstalling Windows was a last step I was about to take.
Luckily I managed to find out the issue. As I've been inspecting drivers and Device Manager constantly, I noticed that beside Vendor Id and Device Id, rest of the data was filled with zeros (revision, susbsytem, etc.)
This gave me a thing to ponder. I've edited .inf file and looked for the Vendor and Dev Id's my card is supposed to have. And I was right, there was no entry regarding it. Even in official Gigabyte drivers package.
Vendor Id: 0x1002
Dev Id: 0x6790
Basing on the info from here:
I've edited C7152570.inf and added 2 lines to it.
1. Under [ATI.Mfg.NTamd64.6.1]:
"%AMD6790.1%" = ati2mtag_R575, PCI\VEN_1002&DEV_6790
2. Under ; Localizable Strings, where the list of labels is
AMD6790.1 = "7900 MOD - AMD Radeon HD 7900 GHz Series"
After saving the file, I ran setup.exe from the package and rebooted. Card was installed properly in Device Manager. I ran the installation of CCC package again in order to get all the functionalities available.
So far, everything works like a charm. Gonna test the stability a bit and let you all know how it behaves.
Question to community:
Is it an error on AMD/Gigabyte behalf that Vendor/Dev Id hasn't been included in .inf file or is it card's bios fault that it has non-existant Vendor/Dev Id?