The 14th Amendment is not about restricting States' power. To say so is to the make the same flatfooted mistake as saying that the Civil War was about States' rights. It's true in a certain sense, but completely missing the point. It also misses the point to say, as some revisionist historians might 2, that the Civil War was caused by economic class tensions. Again, it's true in a certain sense, in the same way as saying that "the colour of the sky during the day is bright" is true, but it is so incomplete as to be irrelevant. The core "State's right" issue behind the Civil War is also the core "economic issue" behind it: slavery. 3 The Civil War was about slavery and whether or not States can rebel against the Federal government. The 14th Amendment is about slavery, and preventing rebellion.
Nominally the 14th Amendment mentions restrictions on States' powers, but read closely, it's about protecting citizens from abuses by the State, in particular slavery, as well as other methods to disenfranchise citizens to make them easier to enslave. It's first section might "restrict States' power", but that's like saying "the sky is not red". It's real purpose is to give rights to the citizens and to protect the citizens from enslavement or political situations that might lead to it. The second section is still about slavery, in particular, it is correcting the blasphemous error that my ancestors are 3/5th of a person. The fourth section nominally pertains to the economic powers of States, but when read in historical context it is clear that this section is meant to ensure that neither the federal government nor any individual State would pay for losses from the ending of slavery.
The outlier here is, of course, the third section, which does not "restrict" the power of any State. Rather, it restricts the rights of persons who have engaged in insurrection. An interpretation of this section that gives States the right to disqualify a person for President is not "at war with the whole thrust of the 14th Amendment". The 14th Amendment is not about restricting States' power, at least if one is not being reductive. It is about protecting the rights of people from the States and protecting the union from rebellion.