I don't know what are the motivation for the formulaton in terms of $C^*$ algebras, but you are essentially asking for hausdorff compact/locally compact spaces such all their compact/locally compact quotients can be embeded into them.
Finite topological space do have this property. But the Hilbert cube also have this property (and the corresponding $C^*$-algebra really don't look approximately finite dimensional ).
Edit: I realize there is a small gap in the previous argument: If you consider non unital $C^*$-algebras of $C(X)$ then they do not correspond to quotients of $X$, but to proper quotients of an open subspace of $X$, so the topological interpretation I gave does not work. This being said, the Hilbert cube is still a valid counter exemple: any separated quotient of an open subspace is metrizable and separable and hence can be embedded in the Hilbert cube.