If I were to use an LCD or OLED, would the St7565 or ssd1306 be found in the display or on the breakout?
Without some further information it is difficult to answer this question with any great certainty.
It will depend what you class as the LCD.
Do you have a part number of product link for a display you are interested in?
http://www.adafruit.com/product/438
It looks like the controller is installed in the breakout board.
