2.8 Touchscreen does not work with Proto Screw Shield
You are set to use the ICSP header for communication with the screen. That is good, when you are directly connected to a Mega.

However that is bad when you are going via a shield that doesn't have an ICSP header.

The ICSP header is the 6-pin (3x2) header in the right of your picture. It has SPI signals that the screen needs. Your screw shield lacks those pins.

You need to pass those signals through. The simplest way would be to add a 3x2 stackable header (or two 3x1 stackable headers) upside down through the correct holes to line up with the ICSP header.