I have no experience with Youtube videos and streaming, but considerable experience with photography. I'd just like to double down on some of the comments made by others. Using a larger sensor, such as full frame or APS-C, automatically gives you a shallower depth of field, which is the opposite of what you want. If depth of field is the major issue you're dealing with, something like MFT, or the even smaller sensor found in a quality point-and-shoot, would give you a greater depth of field. And remember that a 1080P video only has a 2 megapixel frame size, so resolution/sharpness isn't likely to be an issue. But the smaller the sensor, the more light you will need to get qualilty video. Increasing your ISO setting can substitute for increased lighting to some extent, particularly with modern cameras, but the laws of physics eventually get you.
Actually, a quality camcorder in theory should be a pretty good solution - small sensor, good lens, extensive zoom capability. But unfortunately, most camcorders have a lens that's just too long even when fully zoomed out. You have to put them too far away to get the field of view you need.
Something like a GoPro is about the right size, but the lens they have is so wide-angle that you get distortion.
I honestly don't know what the ideal camera would be for this. But so many people are making these videos now that someone should come up with the ideal low-cost camera designed specifically for Youtube videos. It would have a small, high-quality sensor, zoom lens, auto-focus, good near focus, external mic jack, AGC enable/disable, and full swivel display. And I also wonder whether USB output would be more useful than HDMI - in other words, it would appear to be a webcam, with no BlackMagic converter needed, but with no loss of quality.
On a side note, if you can't turn off AGC in the Canon camera, Magic Lantern may let you do that.