I think you can sort-make the argument you want to make, because all computational basis states are also product states, just like all computational basis states are also stabilizer states and also (if they have the right parity) fermionic gaussian states. Where it goes wrong is when you start thinking about gates/circuits instead of states, because (e.g.) CX or CCC..CX gates with arbitrary numbers of controls are (in general) highly entangling gates, but they map computational basis states to other computational basis states.
Does this make sense?
|Simulated state |Computational basis state that allows classical simulation|
|Non entangled superposed state |product/stabilizer/fermionic state|
|Entangled superposed state |stabilizer/fermionic state|
|Magic superposed state |fermionic state|
So as I see it non superposing states are a special case of any classically simulable superposed state so they can always be classically simulated.
For example imagine we have a big, complicated quantum circuit, which doesn't look "nice" in any way, but at every time it happens that the state may be expressed as a superposition of only 2 (or a few) stabilzer states. Even though the states are simple in this sense, we wouldn't expect to be able to efficiently classically simulate this circuit.
This is because even though there is an efficient classical description of the state of the quantum computer at every timestep (the efficient description is as a superposition of a few stabilizer states), there isn't an efficient way to find these efficient classical descriptions, or even notice that they exist.