If I am reading the question correctly. You have the 600d connected to your computer and using the eos utility to remotely control/view the camera. You also have some form of output from the computer, I am assuming hdmi and the 'hd cable' is a typo, connected to your tv. You can view pictures and video fine in full screen, but you are not seeing live view fill your screen. You want to know if buying a full frame camera will change this.
What you have is a resolution problem, not a sensor problem. I can't say that I have specifically used your setup but my first instinct is to check the tv or the eos utility that it isn't sending scaled down information out of your laptop (which from your description sounds like what it's doing). Have you tried going straight from the camera to the tv? I am sure you can set the resolution of the tv to whatever it needs to be to fill the screen. Then using magic lantern turn off the display.
What you have is a resolution problem, not a sensor problem. I can't say that I have specifically used your setup but my first instinct is to check the tv or the eos utility that it isn't sending scaled down information out of your laptop (which from your description sounds like what it's doing). Have you tried going straight from the camera to the tv? I am sure you can set the resolution of the tv to whatever it needs to be to fill the screen. Then using magic lantern turn off the display.
