First, I'd like to understand the basis of the claim that the light leak would cause under exposure. I've heard this many times and it just does not make sense to me. Theoretically, the meter should only measure the light through the lens to give the "correct" exposure for the image in the frame. In reality, the light also "leaks" through the viewfinder and, in the case of 5D III, the top LCD panel as well. My guess is the light leaking through the LCD Is far less than through the viewfinder. I don't think you will get underexposed image if you don't put the viewfinder cover on.
Yes, the light leaking though the LCD can be far smaller than what leaks though the viewfinder. It can also be sufficient to be a source of error if everything is sufficiently dark. For example, the case of a relatively well sealed viewfinder against your eye v. a spotlight shining on you and the camera but not in the lens.
The second part however, largely depends on how bright everything is relatively speaking. If I'm shooting a dimly lit subject from an even dimmer area I don't necessarily have to block the viewfinder, and the backlight will still cause a metering error. Actually, at least according to my testings, so long as the whole environment is dimmer than about EV1 turning on the backlight will alter the meter reading regardless of whether you have lens caps or viewfinder blocked.
A lot of the problem here, IMO, is that there has been lots of "cursory testing", gnashing of teeth, and complaining about how the camera is broken, without understanding either the design limits of the camera or doing any serious controlled testing inside and out of those limits. For example, the original Canon Rumors test with the lens caps on is so far outside the designed limits of the metering system as to be utterly useless in speaking to anything other than to show that there is in fact some light leaking from the LCD/LCD backlight to the meter.
I've conducted more controlled tests, at least to the best of my ability though even I admit that they aren't done to my fullest satisfaction (largely because I don't have sufficient equipment to do so), and they seem to indicate that the meter functions properly inside the designed range (1-20EV) unless you shine considerably brighter light on the top LCD than the lens or viewfinder can observe.
In practice, the situations where you're both operating inside the stated specifications of the meter (1-20EV) and doing something that would put a sufficiently bright light on the top LCD are vanishingly small. Even the assertion that this defect affects astrophotographers and nighttime landscape shooters, runs into problems when in all likelihood what they're metering is already outside the camera's design limits to start with, where they're no guarantee of function anyway. Which is why I keep saying that there is the distinct possibility that Canon will decide that no mitigating action needs to be taken and that the problem only occurs outside the designed limits of the camera. I sincerely hope they don't, since the fix is probably a bit of black tape somewhere, but it's a possibility that can't be ignored.