First, the "state action" requirement is the norm for most civil rights violations. The U.S. Constitution is structured in such a way that it is designed solely to protect people from government overreach. Statutes engaged by legislative bodies at the state and federal level protect people from misconduct by private parties.
For example, ordinary state criminal laws would ordinarily forbid private parties from acting as armed vigilantes to prevent people from doing pretty much anything. And, if armed vigilantes were doing something that government officials and politically powerful people in the state didn't approve of, they would have been arrested, charged and convicted of crimes by the police.
The trouble is that in the U.S. legal system, if the government decides not to enforce its criminal laws, you are limited to bringing civil lawsuits against the offenders for damages or injunctive relief (e.g. a court order ordering the vigilantes to stop misbehaving). But, as a practical matter, private lawsuits are not very effective when the courts and government officials are determined to thwart you and so are a significant share of the members of any jury pool. (This is one of the reasons, as a matter of logic, for the Second Amendment, because you need some way to protect yourself from crimes if the government exercises its right to do nothing to protect you from criminals even when it is capable of doing so.)
The U.S. Constitutional and legal system requires the good faith of the elected officials in state and federal government to work properly. But, if majorities of voters under whatever flawed electoral system is currently in place are bound and determined to suppress a political minority, the democratic features of our political system make it very hard to intervene to protect that political minority.
As you note, the 13th, 14th and 15th Amendments to the U.S. Constitution (collectively, the Civil Rights Amendments) all empower Congress to enact legislation enforcing those provisions in addition to any self-executing effect that they have, although the exact scope of that enforcement power was at the time a matter of some legal debate and in any event often wasn't exercised by Congress (which by then had white, anti-civil rights members in many House and Senate seats). But, in U.S. v. Cruikshank a very racist U.S. Supreme Court held that the state action requirement applied not just to self-executing aspects of the Civil Rights Amendments, but also to the 1870 Enforcement Act adopted by Congress during Reconstruction with an intent to stop precisely the kind of armed private action that occurred. It held that the 1870 Enforcement Act as applied to private parties was unconstitutional.
(Around the same time the Slaughterhouse cases gutted the privileges and immunities clause of the Constitution, and Plessy v. Ferguson gutted the 14th Amendment with the "separate but equal" doctrine.)
However, if the state fails to address these actions, would it not
then become the federal government's jurisdiction since the state
failed to enforce the Fourteenth Amendment within its borders?
No. The U.S. Constitution doesn't provide much protection at all against state inaction. (The human rights tribunal for the Americas has faulted the U.S. for violating human rights by not having protections against this kind of conduct which all Latin American countries do have.)
Would the plaintiff have needed to sue the state for failing to
enforce the Fourteenth Amendment in order for it to be under federal
No. The plaintiff would have no remedy, which was basically the intended and expected effect of U.S. v. Cruikshank. The justice of the U.S. Supreme Court weren't any more stupid then than they are now. They just had different values and part of those values was the belief that it wasn't the place of the federal government to curb the acts of groups like the KKK and Jim Crow laws (in part to prevent a re-ignition of the Civil War in a post-Reconstruction period when white supremacists were more powerful than ever in the states that had previously fought a war with overwhelming support from majorities of people entitled to vote to preserve slavery). The Civil Rights Amendments were shoved down the throats of the Southern states as a condition for readmission to equal standing in the Union free of military law and civil disabilities during Reconstruction. So, they ratified those amendments under duress and then did everything they could to reinstate the pre-civil war system as completely as possible which they had still believed in despite losing the Civil War (and indeed, modern political figures like Roy Moore, a previous Alabama Supreme Court Justice, still want to return their states and the nation to the ante-bellum South, and argue that slavery was as much good as it was bad for the slaves and for the South).
It took a century and two World Wars and the Korean War, in which African-Americans served in the military for the U.S., and a thriving post-war economy, for those attitudes to change.
Ultimately, in light of this ruling, in the 1960s, the Civil Rights Acts were adopted by Congress to regulate discriminatory conduct by private parties, not with the justification and authority of enforcement powers of the Civil Rights Amendments, but instead based upon the interstate commerce clause whose scope had been dramatically expanded during the Great Depression in cases like Wickard v. Filburn, a choice which a pro-civil rights U.S. Supreme Court gave its stamp of approval in the Heart of Atlanta Motel case.