The 14th amendment's insurrection clause specifies Congress as the body which can pardon an insurrectionist and allow that person to hold office. But the clause is vague as to how it is enforced, likely given that it was written in real time and everyone then KNEW who the insurrectionists were. Since it was written in a given time for a given purpose and I doubt the writers even imagined another such insurrection 150 years later, this might be the tack the conservatives use - not the state's rights aspect.
But given it is vague and doesn't even spell out a definition of insurrection, a plain reading of the clause clearly should allow a given state to make their own determination. Plus, there is zero legal precedent that a person barred under the clause would have to have been convicted in a court of law. None of the Confederates barred at the time were convicted.
Insurrection under the clause is as much political as legal