Is it possible to run any sort of hardware diagnostics through ADB
Investigating in this issue, we already have a lot of broken screen questions providing some clues:


You could hook your device to your TV screen via HDMI. This should mirror its (invisible) screen. Of course this would require your screen at least to respond to touch events.
AirDroid might provide some help, as it allows to manage your device for many things, and also advertizes to stream the screen of your Android in nearly real-time
You could use some VNC server, and start it via ADB as described in Broken Screen + Using adb shell to run vnc server


The latter obviously was the solution fitting you best, as you reported having done exactly that with VMLite VNC server. Not the cheapest solution (this app comes for about US$ 8) – but certainly cheaper than paying off a tech :)