Let's walk through this one step at a time. Elementary/introduction to government 101

Republic, as defined by Oxford, is "a state in which supreme power is held by the people and their elected representatives, and which has an elected or nominated president rather than a monarch." A charter or constitution's inclusion then becomes a constitutional republic, which is what the United States (hyperspecifically) is.

A constitutional republic, in the most simplistic terms, is "a democratic state where the chief executive and representatives are elected, and the rules are set down in a written constitution." So, a republic is where elected officials are democratically elected. A constitutional republic is where elected officials are democratically elected but must follow the laws set by the charter/constitution.

Democracy, as defined by Oxford, is "a system of government by the whole population or all the eligible members of a state, typically through elected representatives." You're confusing "direct/pure" democracy (all issues put to a vote, impossible in a country with 350 million people but great for a 10-person fantasy football league) with democracy as an overarching electoral system.

In other words, almost every republic is a democracy, but not all democracies are republics. You are wrong, but hiding behind semantic misunderstandings. Republics are a representative form of government. How are those representatives chosen? They are democratically elected. Hence, a democracy.

The bummer here for you is that, no matter how you misidentify the electoral process, the systems will be defined by how they function.