On a side note, you MAY be able to use a custom firmware to make the 6D suitable (enable 'clean' HDMI output, disable the auto-off).
Google search for 'Canon 6D Magic Lantern'. As it's a custom firmware it will void your warranty and comes with some element of risk, but many people use it to extend the abilities of their Canon DSLRs.
You'll need to check the ML feature list to see if the 6D gains clean HDMI and the ability to disable the auto-off though before you start, otherwise it'll be pointless (aside from making your camera a better camera, that is).