Why is satellite imagery data delivered as signed 16 bit integers?
Because the surface or top-of-the-atmosphere products are not raw data. Atmospheric corrections have been applied to the raw data and those corrections have been done with a precision that can't be properly captured in 8-bit numbers. You can round data outputs to 8-bits, but there would be information loss.
Some bands, like atmospheric opacity can be calculated by the model with high precision, and you need a 16-bit number to store such precise values.
Although the values are stored as 16-bit integers they are floating point numbers, and the scale factor is provided in the same table you mention.

I also think that it have to do with having a coherent data format across all the Landsat derived products, including Landsat 8 that have the OLI sensor with a spectral resolution (or bit depth) of 12-bit, that means that the intensity of each pixel is recorded with a number between 0 and 4095, something that provides much more information than the traditional 8-bit (Used by Landsat 7 and before), were intensity is recorded with values between 0 and 255.
Additional bits give more information and is specially useful to distinguish details in areas with low contrast, like shadows, water or snow fields. For example, all the shades of white in a snow field that corresponded to just one color in Landsat 7, can correspond to 16 different colors in Landsat 8.

Data is distributed in 16-bit files because computers are optimized to deal with 8, 16 or 32 bit numbers, and most software reads images with those bit depths. Therefore, a 12-bit file would not be very compatible, and the smaller standard file that can store all the information of a 12-bit image is a 16-bit data file.

The bands that use 8-bits in the table 6-1 that you mention are quality control bands, that contain information that can be stored in 8 bits without any loss in performance.