The state legislatures would just choose the electors.
Yeah but what if the states don't choose any electors, huh? What if all of the House of Representatives and Senate all resigned and there was no one to vote in either house and also no Speaker of the House and no president pro tempore? What if no one votes in any special election for the Congressional delegation and no governor will appoint any Senators? Then you've got no one in the executive brange, and no one in the legislative branch. All you need is all of the SC Justices to quit and we have done away with the federal government.