Yes, but the Hadamard transform is essentially a linear operation.

The span of qubit states reachable in a polynomial number of Hadamard transform operations from the starting state is much less than all possible qubit states.

This follows immediately from information-theoretic principles. There are 2^(2^N-1) possible 'collections' of 'classical bits' representable by N qubits. Therefore it takes 2^N-1 bits of information to represent, but since the Hadamard transform is basically linear, we can see (in a very handwavy way) that only a polynomial number of possible states is reachable from a polynomial number of quantum gate operations.