On those cases the law codifying and defining Insurrection had not yet been passed, but the Amendment gave them the power to make the decisions to allow or deny ballot access, the congress acted with their authority given in Section 5 of the Amendment. Since that time they have, again within their authority, defined and codified what Insurrection is in the aforementioned USC Section 2383. This makes it so that someone can be convicted of the Federal crime of Insurrection, if they were convicted of this Federal Crime it would meet the aspects of the Amendment 14 Sections 5 and 1 that I have described herein.
All elections are state run elections, not just Primaries, but they do not get to decide who is eligible to run for President, that eligibility is also listed in the Constitution, which gave Congress the authority to act on and to enact laws regarding Amendment 14, which they did later when they codified a definition in law for Insurrection as described in Amendment 14.
Secretaries of State or any authority below the Constitutional guidelines, simply do not have the authority to just "decide" such things. Imagine if they did. The Texas SoS would just decide that allowing border crossing without regard to the law was "Insurrection" as they are trying to "take over the government" by "importing illegal voters"... Suddenly Biden is no longer allowed on their ballot because some partisan hack has decided he just cannot be... Florida follows suit, Alabama, and every state with a republican SoS... magic and feels are simply not due process and they just don't have this authority.
A federal criminal conviction, would probably be enough To disqualify someone from holding office.
It is not, however, the only way an insurrectionist could be disqualified.
A criminal conviction is not a requirement, but it might be one of the ways. Despite what you said earlier in the week.