Suppose the electrons of the hydrogen molecule are in the excited antisymmetric spatial state
Yes, in this case the spin would be less restricted if the electrons were bosons. But antisymmetric spatial states themselves are pretty restrictive in general. The precise sense in which the antisymmetrization requirement is more restrictive than the symmetrization requirement will be explored in the next section.
Since the excited spatial state takes care of the sign change, the spin state should remain unchanged under the electron exchange. That is true for the three triplet states, but not for the singlet state, so the spin state can be any combination of the three triplet states. The complete wave function must therefore be of the form
You can also derive this using the arrow combinations like was done in the text for the ground state. Then the requirements are that