That's virtually meaningless. A "republic" is virtually any country that doesn't have a monarchy or dictator.
So drawing a distinction between a "democracy" and a "democratic republic" in this manner is a waste of time. There plenty of democratic monarchies, which are equally democracies, too.
There has never been a true democracy anywhere and anytime in history, even today.
Every democratic government in existence currently today is severely affected or influenced by monied, corporate, aristocratic, hereditary or powerful interests to some degree. Some countries manage it better than others but all of them fall short of a true democracy ... a system that is controlled by the people and benefits everyone equally.
IMO the US is de-facto like the ancient Roman republic, where plebeians, could vote but only for patricians, so "everybody" (ofc slaves and womens rights were neglected back then) could vote but all questions that were ever discussed in the senate were interests of patricians, same goes for political "coverage" and campaign elections.
So there was dissent and processes that were democratic on the surface but they exclusively revolved around the interests of the patricians.
The US is like that where patrician interests are replaced with capital interests. You can only vote multi-millionaires into the white house and the only issues to ever change are the issues of a fraction within the capitalist class (meaning someone living off of someone elses labour rather than their own).
If you belong to those capitalists you enjoy democratic representation, if not you can only decide which capitalist position you find better and vote for that.
Can't a system be a true democracy and a democratic republic at the same time? I don't see how adding some republic detracts from the democracy.
Republic: "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."
Democracy: "A system of government by the whole population or all the eligible members of a state, typically through elected representatives."
Well they are just different. It's like saying a despotism is a monarchy. Technically a despotism is a monarchy, but having a monarchy doesn't necessarily mean it'll be a despotism.
And it was never designed to be. It was always meant to be a republic.
We first were a confederation. Were your idea of a true democracy was more or less in place. The revolutionary war was won in 1783. The constitution wasn't ratified till 1789, and the bill of rights written until 1793. Before that the US had almost no central government, and each state was independent from one another. Had their own currency, banking system, laws, and military.
States still have a lot of that same autonomy today, but there was no central government tying them together. If the US went to war and a state didn't want to go, they wouldn't. A little more complex than that, but generally that's what it amounted to.
Having this type of system created a bunch of problems and came to a head when Shay's Rebellion happened. I won't go into depth about it, but mainly confederated Massachusetts couldn't fight off the rebels attempting to take over the state. Since the US was a confederation there was no central government the state couldnt call on for help, and all the other states more or less said 'meh sucks for you'.
This incident lead to the Constitutional Convention that wrote the document we still uphold today, and bringing in more of a centralized Federal Republic, and not a decentralized confederated one.
My ranty point is, we tried the whole true democracy thing and it failed. So we went to a Federal Republic, still very much democratic, but moved away from a true democracy.
"republic" is opposite to "monarchy". It is unrelated to democracy or authoritarianism. Nazi Germany was a republic. France is a republic.
Your republic is flawed by design. Your founders didn't trust democracy so they weakened it, the country hasn't managed to improve the democracy since.
Australia is also a Federation, but a monarchy not a republic. Australia is quite a bit more democratic than America