To be perfectly clear America is a Democratic Republic to distinguish it from other type of "Republics". Rome was not really a Democratic Republic although there was voting by the people.
It was a quite complex government.
But traces of what evolved later could still be seen in it. Yes, the aristocratic Senate had most of the power, but then you had the Tribune of the Plebs.
That was a rather powerful position, that was voted on by the plebs themselves (the population). And they had a direct voice in the Concilium Plebis (People's Assembly). And they were a critical check on the power of the Senate through the Tribune of the Plebs. And their powers and abilities were almost sacrosanct, being able to do things like summon the Senate, propose and veto legislation, and veto consuls and magistrates. And not unlike the House of Commons in the UK or the House of Representatives in the US, they held ultimate power over the "purse strings" of the Republic.
It is also in reality what the Emperor became. When Caesar was appointed as Dictator (something very different from the modern interpretation), he was granted the powers of the Tribune of the Plebes. That is why when his nephew Octavian (Augustus) became the first Emperor, one of the things he incorporated into his power was he was also the Tribune of the Plebs.
So it was "Democratic", and much of what was later refined in the UK and refined again in the US can be seen in how their government worked. I actually have long had a fascination with how it all worked, especially the Gracchus Brothers. Who each was elected as Tribune of the Plebs, and proceeded to tie the Senate into knots by vetoing their attempts at grabbing more power. Primarily through the use of their veto, legally dismissing their attempts to do things against the interests of the plebs. Even ordering the treasury locked in order to curb spending.
Julius Caesar wanted to become Tribune of the Plebs, but instead entered the First Triumvirate and ultimately Dictator. However, he did get Marcus Antonius (Mark Antony) to be elected as Tribune of the Plebs. And through him was able to keep the Senate tied up because every time they voted to reverse his reforms, M. Antonius would use his power as Tribune to veto it. And under Augustus, that is the power he and later Emperors actually used in order to rule the Empire.
Which is the key difference between Rome in the Republic and Empire eras. As a Republic, the people voted their Tribune. In the Empire, the Emperor was the Tribune. And ultimately, not unlike what formed in the UK with the House of Lords (Senate) and the House of Commons (Concilium Plebis).