In low-light settings the quality is just not good, blurry. I really just do travel photography so don't have time to mess with settings if I am capturing scenes of people out at night in a busy Chinese pedestrian street, for example.
Then get a good phone that does computational photography very well. It will think for you more than and better than any DSLR on the market. It won't (at least not at this point in time) tell you how to compose an interesting image, though. You've still got to figure out where to place the camera and where to point the camera.
Interchangeable lens cameras are still created with the assumption that the photographer wants at least some control how the image is captured and processed. Like many advanced tools, using them well requires the effort to learn how they can best be utilized. If you're not willing to do that, then get a tool that is designed to guess well, at least most of the time, about what you want the photo to look like.
I am just wondering will I notice an improvement in the quality of my photos if I upgrade from my Rebel T3 to a D7200.
Probably not. If you had used a D7200 instead of a Rebel T3 to take the example image, you might have a slightly less blurry result of a poorly composed image that makes no sense at all. It still would not be a good photo.
In the hands of someone who understands the differences between the two cameras and how those differences may be used to leverage a better photo taken under the same conditions, there probably would be a noticeable difference.
Things happen fast, I'm usually walking with a group and don't feel that I have time to mess around with settings in manual... if the answer to my problem is suck it up and get fast at shooting in manual and you are going to have the same problems with a D7200, I am receptive to that feedback haha.
Better gear, in and of itself, doesn't make anyone a better photographer.
If you want to be able to take better photographs, you need to be willing to learn how to be a better photographer. That begins with learning about composition. It means learning about how the intensity, color, shape, and direction of light affects the way the scene or subject looks. It requires understanding how exposure works and how different intensities of lighting affect the way a camera needs to be set up to get a good exposure. Which camera one uses is way down the list of things one needs to know.¹
The blur is just one complaint I suppose, I also just feel that sometimes my photos don't have the "wow" factor that you would expect from a DSLR, this is not something I can explain further. Wondering if the D7200 having twice the megapixels would help with this.
A good photo isn't about megapixels, or even whether there is a little blur. It's about how does an image tell a story? How do the lines, colors, highlights and shadows, etc. lead the eye from one point to another? What kind of emotional or intellectual response does it invoke in the viewer?
In the end, gear with higher capabilities can certainly help. But a better camera won't make you a better photographer. It will just allow you to use more of the skill, knowledge, and experience you've picked up along the way. Part of that experience and knowledge contributes to the ability to pick the best tool for the job from among the options one has available.
Does the camera matter?
When should I upgrade my camera body?
¹ There are times when being a good photographer does mean recognizing what technical capabilities are needed from the gear to get a desired result. Understanding the capabilities of the various pieces of equipment available and selecting the lens and camera that will give the best result draws from the knowledge and experience of a veteran photographer.