Folks repeatedly say you must be inconspicuous yet it's still a camera up in your grill. If it's not spytech, it's seen, if it's seen, well then the notion it must be rangefinder sized is just fashion for the genre. Shoot with what you got.
Don't have a 1D but I use a gripped 7d on a hand strap which isn't to far off in size. I don't find body size an issue I believe lens size (love those little pancakes) and noise more important. Noise is the worst thing about my combo. So 1dx noise would be my concern.
However, my "street photography" is usually at street fairs and festivals where there is a lot going on. I will usually use a 24-70 2.8 II.
I try to be stealthy, and nonchalant, and do so rather well. If I see someone that is particularly interesting, I may approach and ask permission to photograph him/her while he goes about his business.