In the specific circumstance of a spotlighted singer on a darkened stage then this I would think would be just about the perfect time to be using spot metering. Any sort of metering that is averaging the whole scene is almost certain to overexpose in such extreme lighting conditions. Using evaluative in this case the camera is still going to try to expose for the darker background to some extent, while not totally losing the highlights. The problem is that the camera doesn't really know if it is looking at a dark subject, with a bright highlight in the center of it. Or if it is looking at a bright subject, on a dark background, either situation will produce the same result as far as the metering system is concerned. This would be true for still images, as well as for moving ones, so in this case it's not really a function of using the camera for video causing this issue, but a general photography one.
Of course the issue of appropriate shutter speeds and apertures for video work still apply, where 1/2×fps is the normally accepted shutter speed for natural looking video playback, going longer with the shutter speed is difficult as you need some time for processing the data off the sensor between each frame. Well at least once the frame rate has reached around 24 fps. Below that motion will look choppy whatever you do. With cine film the shutter is usually mechanically coupled to the film transport mechanism. Since the film is generally pulled through the gate, then stopped while the exposure is made, before being moved again for the next shot. The shutter either being a simple rotating disk with a 180° aperture in it, or a spinning prism in higher end cameras. This is why these relative speeds are known as 180 degree shutter. At least things are now better than when I started shooting Standard 8mm cine which IIRC offered choices of 9, 12, or 18 fps. I seem to remember shooting in the middle mostly, to balance quality with running time, especially as with Standard 8 your 50' of film was achieved by running a 25' length of 16mm film through the camera in opposite directions, splitting it in half during processing, and splicing the two halves together. I recall that the full 50' gave a run time of about three and a half minutes. So modern DSLR video, with 60fps, and a single take of 30 minutes seems like real luxury in comparison. With 8mm cine it was an exposure of 1/18, 1/24 or 1/36s, but the Kodachrome cine film we used was only ASA/ISO 40, yes 40 not 400, so the usual problem was getting enough light, not too much, even with an ƒ/1.4 maximum aperture.