True, but simple color contrast can work wonders. It’s not just a symptom of flat design — it’s a symptom of flat and pretty much all the same color design.
It might be a touch ugly, but as an example, if the main interface were black and the URL bar were white, you could nix all the texture/shadow/borders/margins you want and still be fine.
This is why I use the high contrast mode in MacOS. It ads nice solid black borders to all inputs. Originally I tried it out while fixing a bug, but now I can’t go back lest everything looks so monotonous and washed out.