Except in this case I was already wide open (f/2.8 on an f/2.8 lens), and without a flash installed (5D has no popup). Something else is happening.
In Live View or video shooting, there is only contrast detect AF. That becomes even more difficult in low light, or a low exposure. Probably to help the one-shot video AF obtain a focus lock, the algorithm is "if current exposure is below a certain level, momentarily increase ISO". It may also momentarily decrease shutter speed -- I don't know. It does cause a momentary brightening on the LCD, but not always.
Since the 5D3 AF system is designed primarily for stills, this fluctuation has no effect in that shooting mode. In video it does cause a slight disruption, but that was likely not a design priority several years ago when the design was frozen.
The 5D3 AF behavior is obviously not necessary, as other contemporaneous designs like the D800 don't do that. However they don't have perfect video AF either, and for serious DSLR video you typically use manual AF.
Back when the 5D3 was designed (possibly around 2010), the designer's viewpoint may have been a high-end HDSLR is normally not used for casual video, most serious videographers manually focus, and the then-available technology did not permit fast reliable video AF in a shallow DOF situation at an achievable price point. Thus they may have used a somewhat rudimentary single-shot video AF system, and the brightness variation during video AF was a stopgap to improve AF lock performance. I'm just guessing.
Time and technology march on, and it's likely the 5D Mark IV will have a greatly improved video AF system. Newer cameras like the Lumix GH4 and Canon 70D show improved video AF is possible in DSLR or similar cameras. However those cameras (even though newer) don't have the low-light performance of the 5D3.
The problem is at that market segment you have to deliver superb low light performance, and anything you put on the sensor like phase detect pixels could theoretically degrade that. There are also issues with legacy lenses and how well they can be made to work for video AF. E.g, the 70D or Lumix GH4 only produce best video AF performance with certain lenses.