When a battery goes flat it has less power, so it is reasonable to expect things to slow down. To maintain the same speed on less voltage you would need more current, that could give serious problems. Sounds like a safety feature to me. Bigger batteries would mean a heavier camera.
I don't buy that explanation. I mean, if the battery is flat, yes, but 50% is still plenty of power.
I don't say the phenomenon don't exist, because that I can believe. But it being direct affect from the battery level, I wouldn't think so. I've worked plenty on the cell phone designs, and the batteries we have usually work around same characteristics, ~1500-2000 mAh Li-ions. Cell phones are 3.7V, and 5D3 at 7.2V, but that's just because they use 2 cells in series.
Until about <25% of the capacity, the voltage level still stays considerably high, and there's no big voltage dip due to bigger current draws either. And cell phones draw >1A when they get power hungry. I don't know the 5D3 power draw when everything is active (taking photo, DIGIC crunching data as fast as possible, memory cards writing, mirror and shutter going, etc.) but I'd be surprised if the peaks are more than 0.2A (not including the LCD operation, that could take 0.1-0.2A alone). (if someone knows real current draws, please enlighten me).
Assuming that's true, which I don't know, the battery level could go down to <10% and still the voltage dips, caused by the current draw, would be minimal.
I haven't seen the schematics for Canon, but I'd be also really surprised if they don't have a regulator at the input. In my world that would be just poor design. If they do have regulator, it would eliminate the actual battery voltage completely at the camera end, and until battery is really down to <1%, the camera would receive proper voltage/current. Again, if someone knows the design or has the schematics, I'd be happy to learn the details.
I have the 5D3 with grip, and 2 Canon batteries. Occasionally I'm down to ~50% on long day, but I don't remember seeing this phenomenon, then again, I wasn't looking for it. Next time I'll try and see.