While swapping lenses is a hassle sometimes, I suspect carrying two cameras would be even clumsier and makes your shooting experience less enjoyable. How about the weight? You may potentially solve one problem while creating another. Maybe you could borrow any camera from a friend and see how it feels carrying two cameras on a shoot. If it works for you, get the 6D for shooting portraits and landscape. Personally I think only wedding photographers must carry two cameras. That's just my opinion.
I think practice doesn't support your opinion.
First, there are very sophisticated systems around for carrying two cameras. And personally, I've never understood the "extra weight" argument. If carrying three of four extra pounds for a few hours is too much, perhaps you need to lose some weight. Lose five pounds and then you'll never notice the second camera you're carrying.
Most photojournalists I know and see customarily carry two bodies -- usually one with a wide lens and one with a 70-200 or similar.
Experienced event shooters often carry two bodies, sometimes one with flash, the other without.
I shot a surprise birthday party recently and without two bodies I could not have simultaneously captured both the crowd reactions and the celebrant coming in the door -- ultrawide on a tripod aimed at the crowd (remote triggered) and a 35mm on the other. Then for the rest of the event I switched to a 35mm with flash for candid and posed group shots and a 135 for unobtrusive, longer and more personal portraits types.
Except for street photography or just out walking around, I'm using two bodies most of the time these days.
It's a lot more versatile and much easier and more convenient than constantly swapping lenses.