Which case ultimately led to the conclusion that the Supreme Court could decide on state election matters in the case of ballot disputes?

The case that established the Supreme Court's authority to intervene in state election matters, specifically regarding ballot disputes, is Bush v. Gore. This landmark decision arose from the contentious 2000 presidential election where the results in Florida were disputed due to issues with the vote counting process, including concerns over "hanging chads" on ballots.

In the ruling, the Supreme Court determined that the Florida Supreme Court's method of recounting votes violated the Equal Protection Clause of the Fourteenth Amendment because it created unequal standards of counting among the votes. By deciding this case, the Supreme Court asserted its role in not only interpreting federal election law but also influencing how states conduct their elections, thus affirming its power over state election matters when they intersect with federal constitutional rights.

The other cases listed do handle important constitutional issues related to elections and campaign finance but do not specifically address the Supreme Court's role in resolving state election disputes in the same way as Bush v. Gore.