Are Cognitive Load and Willpower drawn from the same pool?

Bingo. There is a large body of literature on the dual-process theory of mind, and the heuristics and biases literature is full of examples where cognitive load interferes with "rational" responding.

I find the second link really interesting. If I interpret it right, the core idea is that in an ancestral environ have-to and want-to are nowhere nearly as opposed as today having to do boring admin tasks vs. screwing around on Reddit. For an animal, almost everything is they do is have-to (increases fitness) and almost everything is want-to too, either interesting or terrifying but boring is it not. For an animal, the closest thing to boredom may be doing a task without any immediate reward, which is kind of rare, a good example would be for an ape to try to crack a hard nut. In this case getting bored with it and stopping is have-to, going after an easier nut to crack increases its fitness, so the change to exploration mode happens and want-to and have-to goes hand in hand.

It takes a human mind to do have-to without want-to, to do a task without any sort of immediate reward, while the animal brain underneath is saying "stop it! you are not achieving anything! you are typing for an hour now and nobody gave you even a peanut!" and this gets interpreted as boredom. So in humans, in sophonts, have to and want to are separated. But the have-to boring tasks are tasks of which the underlying animal brain is sure that they are not working, because there is no immediate reward and thus on the animal level they are fruitless.

This gives us a dilemma. Obviously enough, one thing that would work is rewarding ourselves, the same way animals find an activity rewarding: snacks. Eating a snack during doing admin every hour and not eating that snack any other time would train our animal mind to not be frustrated with filling forms but see it as an elaborate way to mine tasty date fruits.

However it seems also that this quite simply reduces our ability to function as a human. Seeking immediate rewards is an animal thing and the most distinguishing feature of humnans is the ability to invest a lot of work without immediate rewards - sometimes without any rewards at all, like just working towards a goal that comes from an ethical principle. Not needing immediate rewards is the flip side of intelligence: it does not take any intelligence to keep pulling that lever if it reliably makes biscuits come out of the machine: intelligence kicks in when the machine stops working and we need to fix it. And intelligence is an inherently human trait that should not be trained out of ourselves.

How would you do that?

Cognitive load is a fairly general phenomenon, not limited to memorizing long numbers. You can experience cognitive load simply by living a busy life where you are constantly thinking about other things, and not able to be "present".