The lower “U” is a quarter wave matching section. The upper section is a half wave, and in the case of a slim jim, it is a half wave consisting of two conductors, shorted at the top.
EDIT: With my pedant’s hat on, it isn’t a folded dipole, rather a folded half-wave monopole.
The feed point is at the lower end of one of the conductors.
I wouldn’t dismiss the design either, as you say, it is widely used and is well suited to “roll up” wire construction for portable work. Significantly better than a rubber duck on a handheld.
Currently, though (dried fruit again), I’m inclined to think that it won’t be improved by the addition of a common mode choke on the feeder, and might actually work better without one.
I might try some comparison tests, though I suspect the difference will be difficult to measure, and likely to vary in practice.