I agree with your point at the end. The question of what keeps many suicidal people alive is quite interesting, and no doubt complex. I just feel like the introductory question of "Why do so few humans kill themselves?" is backwards: the basic strangeness of human suicide is why we do it at all. Beyond that, there is then the interesting sub-question of why some people feel a desire to do it, but refrain.
This is a fair summary. I suspect that the author might disagree about how strange the basic urge to suicide is, but I completely agree that the really interesting category is "internally suicidal but not acting on it".
The View From Hell is written by an anti-natalist and, if I recall, 'nonpracticing suicide'. This, I think, is someone who has a radically different baseline for "desire to live" than a lot of the people responding to it.
And, on the flip side, a statement like "the drive to survive is what's fundamental" might well sound completely bizarre to the writer. The idea that despair and sadness are about avoiding things detrimental to survival is at best situationally true, as evidenced by all the times when people find those feelings themselves detrimental to survival.
But aside from all of that, I think this doesn't invalidate the basic claim that a lot of the things stopping somewhat-suicidal people are not the desire to live. At the point where someone is considering suicide, their "life drive" wiring is obviously not a whole answer - so it remains relevant to look at the large number of suicide ideators who are held back by fear, or motivation issues, or other indirect reservations.