Just do not slide any cold flash foot, or any continuos light cold foot, into the camera hot shoe.

Well, I am not sure about that.
For example this one is supposed to work fine - http://www.adorama.com/FADSLRLLK.html
It has no contacts on the shoe, just a piece of plastic, can't harm the shoe I guess.

I slide a few gizmos (which I have made) into my camera’s hot-shoes.
There are two cautions I address, the first is electrical conductivity / shorting contacts etc, and in this regard I have no problem with plastic feet: and that is what I use – except when I use paddle pop sticks.
The second is physical damage – As I understand the Canon (and I assume Nikon and most other brands) dedicated Flash heads’ FEET are designed to BREAK OFF should there be a shearing force: for example, if the flash head is bumped from any angle.
The base of the Flash is apparently easily (and inexpensively?) repairable – the top of the camera not so.
I have not mounted a video light on an DSLR: but if I did I would likely use a quick release bracket: for manoeuvrability of the light’s direction, as the main reason.

WW

When I've seen people use the 5DMKII as a video recorder primarily, they used some really funky brackets with it, and then that's where the video light went. Some had steady cam attachments/brackets as well.

In the old days "cold shoe" or "cold foot" was made of metal and was causing short circuit if interfaced with "hot" shoe or foot.

"hot shoe" or "hot foot" would be the one that produces or expects to conduct electric signals.

Plastic shoe is neither hot nor cold, in the old terminology, and therefore is neutral and safe to use in any of hot or cold attachments, as long as it does not conduct any electricity (plastic cannot).

Great question and I came across a great product just recently that may very well fit the bill
they are

Sima SL-20LX Ultra Bright Video Light (Black)

LED's about 3x3" across use an internal battery charge (must be charged from a wall outlet)
provide excellent light, I don't do video, but having a tiny hold in your palm sized monolight that lasts for 1 hour of continuous light is more than worth the $30 it costs
I liked it so much I bought two and they come with a simple screw down adjustor for placing in the hot shoes of an SLR or a video camera, the material is plastic so there is not connectivity issue or messing with the camera's electronics. Find them on Amazon or B&H
great light really