If a repair shop confirmed that the screen was not the issue (did they try another one?), then your issue is on the logic board. I'm a little surprised that the the shop was not able to tell you if your problem is backlight related though as that is a simple test.
Does iTunes recognize your phone? If so, plug your phone into an iTunes enabled computer and shine a bright flashlight on the screen. If you see a dim image, then you have backlight issue. If not, then it could be an LCD driver issue.
If iTunes does not recognize your phone, then it could be a wide range of issues. Ideally, you will need to have this phone looked at by a micro-solderer so that it can be properly diagnosed.
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