A phone that is vibrating with no display is telling you, "My display isn't working, my display isn't working!"
This can sometimes be a software problem—if the phone has never been opened, for example. A hard reset can solve this problem. Hold the power and home button for 10 seconds.
If it's not a software problem, then it could be damage to the flexes of the screen itself—the LCD flex. Replacing the screen can solve the problem.
If not that, then it could be damage to the LCD or digitizer connector on the board. Anyone who has tried three or four screens and then it magically works again—they were probably able to finally clear debris out of their connector or straighten a bent pin. If the connector is damaged, it can be replaced as a board-level repair
If not that, then it can be a blown backlight filter (like a fuse). This can easily happen when using certain replacement screens that have an exposed backlight flex solder joint. Pressing the screen into the frame will cause the exposed joints to short the LCD line to ground and burn the filter. This can also be fixed as a board-level repair.
If not that—you could have drop damage or latent water damage that is causing a problem in the backlight line elsewhere.