Yeah, it's one thing if your commute has no traffic. If you're sitting in traffic on the ZX6R, it's a lot of shifting your body around to get to the ground, get back into position when rolling, etc. The ergos aren't great.
That said, I get the knock on cheap suspension, too. Many of the naked bikes do skimp on suspension components, and that affects how I ride, even if I'm not on the track. I don't like a soft/spongy front fork.
If you're not enjoying the ZX6R, do check out other bikes. Make sure you test ride, if you can, both a faired upright bike (Ninja 1000, Ninja 650, Hypermotard with windshield, DL650) AND a naked bike that's similar (Z1000, Z900, Hypermotard, SV650, Street Triple). You might find you like the fairing.
Never ride faster than you can stop