Why can't Antoine's necklace fall apart?

if we remove a segment of one of the tori the object still cannot be separated by a sphere, and yet can fall apart macroscopically.

Well, sure it can. The key observation is that "sphere" means "homeomorphic image of a sphere" in this context.

Turn the ring with the part removed (the "C") and pull the in tact ring (the "O") through the missing piece. Then you have two separate components, the C and the O, which it is easy to see could be placed inside and outside of a sphere. Let's put the C on the inside and the O on the outside.

Now, think about shrinking that sphere very tightly (but not touching) around the C, then putting the C back where it was. You've separated the two by the homeomorphic image of a sphere.

Think about if we tried to do this with two Os, like your first figure. If the Os weren't interlocked, sure, it's easy, just like when the C and the O were separated. But if two Os are interlocked, there isn't any way to fit a sphere around one of them in the same way as you could with the C. And if you can't do it with only two Os, you certainly can't do it with Antoine's Necklace. Thus, it will stay together.

The necklace is a topological space $X$, together with a natural embedding $i\colon X\to \Bbb R^3$. Since the verb "to fall apart" describes a process, it may best be described by adding time as a variable. So we ask whether there is a homotopy $H\colon X\times[0,1]\to\Bbb R^3$ such that $H(\cdot,0)=i$ and $H(\cdot1)$ is a embedding $X\to\Bbb R^3$ that is in an obvious fashion separated, say a non-empty part of (the image of) $X$ is in the $z>0$ region, a non-empty part in the $z<0$ region, but nothing in the $z=0$ hyperplane. Instead of a separating plane, a separating sphere would certainly also count (and in fact be slightly more general: two concentric spheres cannot fall apart in the first sense, but can in the second; both interpretations have a point as the inner sphere cannot really fall out, but it is not really linked to the outer sphere either ...)

I am unaware whether the referenced authors took this homotopy-and-sphere approach and you overread the homotopy part, or perhaps whether they had some argument that the sphere property of $i$ is sufficient in the given situation (but not in general, as your examples show).