I can't imagine these signals would be so sensitive to solder joint shape
Solder can completely cover the joint and still not make electrical contact due to oxides or other impurities that prevent the molten solder from 'wetting' the surface. A good joint shape indicates good wetting. A shiny surface indicates proper solidification. Joints with a frosted appearance ('cold' or 'disturbed' joints) are full of stress fractures that are often enough to prevent conductivity.
Probing bad joints is inconclusive because the pressure of the probe often makes them conductive. That's why we judge joints by their appearance.
Are you running the Hello World sketch from the Adafruit-RGB-LCD-Shield library? Do the buttons make the backlight color change?