So the solution is to stay on a legacy kernel, with the security risks it entails, while my Windows 10 can perfectly make use of the DX 11 drivers.
AMD's low-middle tier hardware is known to be mediocre, their drivers are known to be poor, and they're also known to drop support for hardware much faster than their competitors.
Yeah, you can run Linux on nearly anything.. but that doesn't mean you should try to sit in front of an old machine running old low quality components from manufacturers that don't support their low margin products.
You never did give your graphics card model number, but according to this page the Brazos platform had two codename variants for laptops and notebooks: Ontario and Zacate. Hondo and Desna were exclusively for tablets. There were no variants for desktops.
According to this page that puts your card somewhere in the Radion HD 6xxx or HD7xxx driver set, and the only references to Ontario are the Radeon HD 6290 and the Radeon HD 7340.
If you go to the AMD drivers download page you'll discover that both of these cards have dropped off the bottom of the list of supported cards in their respective driver categories.
Now hey, maybe your card is a slightly newer model and it's still in that download list. I don't know, since I don't know your exact card model. But my 10 seconds of research says that actually it's probably not supported any more. If I'm wrong about that then you're pretty god damn close to the cliff edge at this point.