Except that many people don't use their DSLR that way and just want better image quality than compacts. There is nothing wrong with that.
Fast lenses are good for DOF control and capturing moving objects.
Moving objects: image stabilization will not save you, you need to crank that shutter speed up. This means either flash, ISO, or aperture, and sometimes the first two don't make sense.
DOF: It's difficult to get someone's eyes-only in focus especially with a consumer DSLR, DOF preview or not. I think DOF is way overused, though, especially by rookies who are all like "omg, I can do something you can't do with most compacts, so now I'm going to use it for EVERY PHOTO even if it's not appropriate!"
For nonmoving objects if you are using fast lenses for better low-light INSTEAD of a flash, you may want to consider getting an external bounceflash, preferably with wireless ability so you can place it appropriately. For nonmoving objects, a kit lens can actually match a prime in low light due to image stabilization and how primes often do not have it--and let you keep the additional DOF which is often a good thing.
For people who like prime-like performance but still retain zoom, the Tamron 17-50 without stabilization is sharp and cheap, unfortunately also a little cheaply-made and slightly noisy. The stabilized version isn't as sharp, so get the Sigma 17-50 OS if you want it stabilized. They are all f/2.8 constant max aperture which is often good enough in terms of DOF, plus have zoom. It's possible that this kind of short-range zoom is ideal for OP who did after all want a walkaround lens.
IMO you're going about things all backwards. You should have kept your first body, and bought more lenses first. The D7000 is a beast, you need a great lens for it (and I'm not a Nikon person, but probably they have upgraded the kit lenses since you first bought them, so maybe you're missing out on a bit of image quality and/or VR, in which case it would be helpful just to upgrade to the current version of the kit lens?)
Yes, you need a prime. 35mm f/1.8 for general use, 50mm f/1.8 for portraits. These lenses will change the way you shoot. You will be able to shoot indoors, in dark rooms, without flash. You will be able to shoot a portrait with the person's eyes in perfect focus, and everything behind their ears out of focus.
Other good options would be something like a 17-55 f/2.8.
The "superzoom" lenses you talk about will only give you the convenience of not having to switch out lenses. You will not gain any image quality or be able to do anything with those lenses that you can't do with your current lenses. IMO you should buy a couple of primes that will enable you to do things that you *can't* do with your current lenses... like shoot at f/1.8.