It is true that the solar flare will induce current in any exposed wire. However the following factors increase the current. The length of the wire, the thicker the wire, how poorly insulated it is, and the stronger the magnetic surges caused by the flare.
Long distance transmission lines are very long, fairly thick, and very exposed. The result is that the surges that they experience are several orders of magnitude stronger than, say, inside of your washing machine or a transformer unconnected to the wire. That surge doesn't hurt the wire, but it can blow the transformer when it hits it.
And hence the problem is mostly in blown transformers. If you have other transformers on hand, you swap them in and most of the system should be up.