Do we fly full size aircraft with video game style joysticks?
They are used, they are just bigger joy sticks than used for RC flying.  

A current example is the Airbus A320 (and related family) of passenger aircraft.  A short video of the joystick in action(called "side stick", start at about 1 minute in).  There is no direct mechanical linkage between the stick and the flight controls.  


(For the A330, but basically the same). 

For more detail, you can download this (slightly out of date, but with enough information to be useful) Flight Controls lesson for the A320 here. 


  I am more curious about thumb controlled flying as you could control
  everything from what essentially looks like a video game controller.


@Rory Alsop noted that ... 


  ... even video game controllers are terrible for this. The bigger the stick the better fine control you have over it. This is why for flight
  sims for PC's and consoles, keen players buy proper joysticks -
  otherwise they are twitchy and hard to handle


Video game controllers require two hands, have poor sensory feedback (as compared to a stick type controller) and are not as ergonomically effective as a multi-axis stick.  Also, there are some "push buttons" on a control stick that allow moving a finger or a thumb to activate other functions, like the "take control" button in the side stick(For swapping control between left seat and right seat pilots).  On some sticks you'll also have the "push to talk button" for the radio, as has been done for decades.  It's an efficient use of the hand, without having to move the hand for multiple tasks.