amongst random length-k (k>2) sequences of independent coin tosses with at least one heads before toss k, the expected proportion of (heads after heads)/(tosses after heads) is less than 1/2.

Does this need to be k>3? Checking this for k=3 yields 6 sequences in which there is at least one head before toss 3. In these sequences there are 4 heads-after-heads out of 8 tosses-after-heads, which is exactly 1/2.

Edit: Ah, I see this is more like a game score than a proportion. Two "scores" of 1 and one "score" of 1/2 out of the 6 equally likely conditional sequences.

This hasn't stood up to replication, as you guess. I didn't follow closely, so I'm not sure of the details. Physical energy is certainly limited and affects mental function efficiency. But thinking hard doesnt use measurably more energy than just thinking last I knew. Which sounds at least a little wrong given the subjective experience of really focusing on some types of mental work. I'd guess a small real effect in some circumstances.

Anything written 20 years ago in cognitive science is now antique and needs to be compared against recent research.

It doesn't seem to be true in any literal, biological sense.  I don't recall that they had concrete cites for a lot of their theses, and this one doesn't even define the units of measurement (yes, glucose, but no comments about milligrams of glucose per decision-second or anything), so I don't know how you would really test it.  I don't think they claim that it's actually fixed over time, nor that you can't increase it with practice and intent.

However, like much of the book, it rings true in a "useful model in many circumstances" sense.  You're absolutely right that arranging things to require low-momentary-willpower can make life less stressful and probably get you more aligned with your long-term intent.  A common mechanism for this, as the book mentions, is to remove or reduce the momentary decision elements by arranging things in advance so they're near-automatic.  

an idea that sounds huge if true: that the amount of motivation one can exert in a day is limited

This is basically the theory that willpower (and ego) are expendable resources that get depleted over time. This was widely agreed upon in the earlier days of LessWrong (although with a fair deal of pushback and associated commentary), particularly as a result of Scott Alexander and his writings (example illustration of his book review on this topic here) on this and related topics such as akrasia and procrastination-beating (which, as you can see from the history of posts containing this tag, were really popular on LW about 12-15 years ago). This 2018 post by lionhearted explicitly mentions how discussions about akrasia were far more prevalent on the site around 2011, and Qiaochu's comments (1, 2) give plausible explanations for why these stopped being so common.

I agree with Seth Herd below that this appears not to have survived the replication crisis. This is the major study that sparked general skepticism of the idea, with this associated article in Slate (both from 2016). That being said, I am not aware of other, more specific or recent details on this matter. Perhaps this 2021 Scott post over on ACX (which confirms that "[these willpower depletion] key results have failed to replicate, and people who know more about glucose physiology say it makes no theoretical sense") and this 2019 question by Eli Tyre might be relevant here.