TFT LCD Tear-down Silver Rectangle Question
That "silver rectangle" is the LCD driver chip.

The silver color is the back of the chip-die which is made of silicon

The function of that LCD driver chip is to take some input signals and drive many outputs which connect to the LCD.

Without the LCD driver all these lines would have to be connected over the connector making that connection unreliable as each line and connection point would need to be very small or you would need a very large connector.

Here's another example of an LCD driver chip:



where it can be clearly seen where there are some inputs but many outputs (note the thins traces).

The technology used to mount such a chip on a flexible PCB is called "chip on flex".