In what way is your republic "representative" then?
I have a degree in political science, and I've never seen anyone who knows what they are talking about use the term "representative republic". The US is, in theoretical structure, both a federal republic and a representative democracy, albeit one with very unequal representation. In practice, it tends to work as something like a plutocratic aristocracy, but its considered impolite in many circles to the point that out.
> I also only get to pick between 2 old white guys every 4 years
That's only true if you choose to restrict yourself to participation in only Presidential general elections, and not all the elections for other offices or earlier parts of the cycle (and also requires an unusually broad definition of "white" [e.g., including Barack Obama, 2008 & 2012] and "guy", [Hillary Clinton, 2016] and "old" [prior to Trump, the last three presidents were elected at ages 46, 54, and 47--though in the first and last cases their opponents were significantly older--which is certainly middle-aged but hardly "old" for a head of government.])
In a democracy I would actually be able to vote on issues and have my voice heard. That's clearly not happening in the USA.