The question of a centre-right government, or a government one does't like, goes to the heart of statehood, doesn't it? I mean that those seeking self-determination must be prepared to accept unpleasant governments. In the early history of Irish independence, there was much conflict over the Free State, rejected by some Republicans. But you could say that Ireland became a stable state, when left-wing Republicans accepted a Fine Gael government, which descended from the pro-Treaty forces at the time of the Free State.
Thus an independent Scotland might see right-wing governments.