The government of the United States is technically not a democracy; it is a republic. We elect people to represent us, but we otherwise have no direct role in government.
The democracy of ancient Athens, however, was a true democracy. The assembly (ekklesia) basically consisted of all the adult male citizens who bothered to show up that day. The assembly was the Athenian legislature, responsible for enacting laws, declaring war, and bestowing citizenship. The council (boule) was like the executive branch; they carried out the laws passed by the assembly, but also decided what proposals the assembly would deliberate. The members of the council were chosen by lottery. Trials were all decided by juries of one's peers, again selected by lottery.
The setup worked fine for the ancient city-state, but wasn't practicable when the U.S. was founded. A Georgia farmer couldn't be expected to ride north for the capital every time the American Assembly met. Even today, flying or driving to Washington would be onerous for the average citizen. But now, we don't have to be there physically when we could be there virtually.
So how do we adapt Athenian democracy to fit modern times? The complex bureaucracy necessary for modern governing poses an obstacle. These positions require a technical expertise that we cannot expect every citizen to have. So let's retain the executive branch essentially as it is.
But let's completely explode the legislative branch. Every morning, a daily boule of 100 people will be selected randomly. (Or whatever number and frequency is most sensible.) Each person, if they wish, will be able to propose one law or proposition. They can come up with their own or use a suggestion from citizens who have not yet been chosen for the boule. The proposal can be as simple or as complex as the person wants.
Over the course of the day, every adult citizen can debate and vote on the boule's proposals. Comments can be shown with the proposal and top-rated ones can be shown first like on YouTube. By the evening, the votes will be counted and proposals with a simple majority pass.
It is then up to the executive branch to turn the laypersons' proposals into reality. If people disagree with the actions of those in the executive branch, then a member of the boule could call for a vote of no-confidence. If that official does not retain majority support, they will hold that office as a caretaker for a month while others campaign for it. All Cabinet-level positions will be decided this way.
An independent judiciary will be necessary to check the "tyranny of the majority," i.e. to prevent our assembly from depriving rights from a minority. For this reason, judges and justices cannot be removed by no-confidence votes. Whether they will be elected by the assembly or appointed by the president and confirmed by the assembly...well, I'm not sure which is better.
And what of that de Tocqueville quote? The American Republic will endure until the day Congress discovers that it can bribe the public with the public's money. Our system cuts out the middleman. If given more direct power over our government, will we use it responsibly?
Well, how responsibly is it used now? Senate Democrats don't want to propose a budget because "The Ryan budget is being so good for Democrats. Why mess that up?" Republicans are playing with fire on the debt ceiling due to their unwillingness to even hear the words 'tax increase.' Regardless of whether this is temporary posturing or permanent intransigence, both Congressional Democrats and Republicans are more interested in playing politics than, you know, actually governing.
Plus, it will be harder for moneyed interests to buy off members of Congress when essentially every American is a member of Congress. Granted, they can still use the mass media to sway the public, but that strikes me as more difficult and maybe even fairer.
But maybe you still have your doubts. So let's get together 10,000 people or so, a good cross-section of America, and do a practice run. After a year, compare the course our fake assembly has taken the country with that of Congress. And then we will have our answer.