Dave said in the video that the area under the curve "represents" the energy in the battery. No one claimed that the area "is" the energy in the battery.
Really?
We're going to get all philosophical about this? Well, okay...
In your mind, do you have a precise notion of how your area under the curve is meant to "represent" an energy? For example, is there some quantitative relationship between the area and the energy. Alternatively do you have a qualitative notion in mind? Or do you mean that the area bears no particular relationship to the energy at all, and we, by convention, are simply using it as a symbol of the energy?
Also he mentions that is a "common technique in the industry to compare areas under the curve on 0.8V cutout graphs"
Until we know precisely what you mean by an area "representing" an energy, there is no basis to decide whether it is reasonable to compare the areas, or even whether such a comparison is "good enough" for your application.
For example, if you claim that areas are merely symbolic of energies then I can learn nothing by comparing the areas. In the same way, I learn nothing about the differences between cats and dogs by comparing the symbols "cat" and "dog".
Remove enough meaning from the areas and it's time to ask whether it's worth thinking about them at all.
What does that mean? well if the battery is considered to be fully discharged at 0.8V using the graph you provided, the ballpark comparison gives you 50% of energy remaining, sure there is an error from the actual 41.(6)% but that is good enough for government work and far away from your "computed" absolute area capacity of 25%
Ok, if you are happy with that approximation, that's fine. However, that approximation has nothing to do with any areas, so why do anything with the areas at all? Talking about the areas would just be a meaningless ritual.
(Of course, I believe that by treating the areas seriously, and working with the right ones, you can get answers that are even better than those good enough for government work. You might even be able to get a job in private industry!)
Edit: the discharge curve on your graph, happens to be the same as the percentage of energy used. So a straight line will do.
The percentage of energy dissipated is quadratic in time (recall that my graph was representing a constant current draw), the discharge curve on my graph is not quadratic in time. The service time remaining is a straight line with respect to time, but that's obvious.