I see there are many questions on Hall subgroups, but I can't find one that answers my question.
Let $G$ be a finite group. A subgroup $H<G$ is a Hall $\pi$-subgroup if its order is the product of some set of primes $\pi$ (with multiplicity), and $|H|$ and $[G:H]$ are relatively prime. Any such order of $|H|$ is a Hall factor.
Philip Hall proved that a finite group is solvable if and only if the group has subgroups of every Hall factor order. The "$\Rightarrow$" direction is proved in Hungerford, but "$\Leftarrow$" was deemed beyond the scope of that book.
This result sees quite deep, in that this characterization of solvable group seems to impose a lot less structure than does the definition of solvable group. From the definition of having a derived series that terminates trivially, a solvable group can be viewed as having abelian factors (quotient groups of the derived series) that can be assembled in a nonabelian way (via semidirect products and Frattini extensions). But at least to me, these two characterizations seem miles apart.
Consider another direction. Finite abelian groups are direct products of cyclic groups. They contain subgroups of every order that divides the group, and every subgroup is normal in every containing subgroup. This is very beautiful structure.
Nilpotent groups ("almost abelian") can be characterized as being isomorphic to the direct products of their Sylow subgroups. Now Sylow groups are $p$-groups, and these have very nice structure. So again we get that nilpotent groups have subgroups of every order that divides the group, but not every subgroup is normal (though there are many normality relationships). We still have lots of nice structure.
But in light of Hall's Theorem, we have that solvable groups need only satisfy what seem to be rather weak divisibility conditions. This seems quite a leap. The only intuition I have been able to grasp is that many Hall subgroups are relatively "large", as many of them have order equal to the product of all but 1 or 2 primes. But I am at a loss as to how this connects to the definition of solvability.
So to the actual question: Is there any intuition as to how the definition of solvable group connects to the Hall Theorem characterization of solvable group?