Let us look mainly at the US, because the US is the best known representative democracy.
One problem with representative democracy is that leaves millions unrepresented; all those who voted the losing party.
This happens because in representative democracy, the politicians have all the executive and legislative power.
It is logical that to acquire that power, political parties fight bitter campaigns. The idea is to discredit, even paint as illegitimate, the other parties.
One of the effects of the vitriolic campaigns of the parties, is that they turn their followers into people who despise and hate the followers of other parties.
In a representative democracy it then becomes impossible for those supporting the losing parties to feel represented.
How could the people who voted for Hillary Clinton feel represented by Trump? How could the people who voted for Trump feel represented by Biden.
The truth, the sad truth, is that millions of voters consider those who vote for other parties as deranged, idiots, selfish, etc.
In other countries the polarisation is less known, but it is obvious that voters in Canada, the UK, France, etc., no not feel too different from American voters.
It is then clear that, today, representative democracy leaves close to half the country unrepresented.
Even in countries with proportional representation, the situation is essentially the same; sure, in proportional representation, voters that in “first past the post” system, would not even elect a parliamentarian, they elect a few in proportional representation, but still have zero representation in the executive.
Even if they elect a few parliamentarians it does not count for much, except if, as it happens occasionally, those parliamentarians happen to hold the balance of power.
The other problem with representative democracy is that it is not really democratic.
Democracy means “rule by the people”. In representative democracies, the people vote, but do not rule. This is the way it is in the US and in all other representative democracies.
In representative democracies, the elected politicians in the executive and the legislative, together with the Supreme Court judges they appoint, rule, the people do not rule at all.
This is why the alienation of millions from representative democracy keeps growing. Some people ate even talking about “the Chinese model of government” as an “alternative to democracy”; in other words, the alternative is dictatorship, a dictatorship that accepts Capitalism, as long as the capitalists (and the million of workers who work for them), accept that those holding political power have all the power; it is not very different from Hitler,s Germany.
This is why we must turn to direct democracy. Direct democracy turns the tables on politicians; instead of them holding all the power, the people have all the decision-making power.
In a direct democracy, a small number of people can put almost any issue before the voters for the voters to decide.
Ordinary voters can stop any law the politicians make. Ordinary voters, not the politicians, can change the constitution, etc.
This power of the people diminishes the power of politicians, as a result they, and the lobbies who support then, and even make and break politicians, see there is no sense in vicious electoral fighting because they will not be able to do anything not supported by the people.
So, direct democracy is more democratic because is is closer to “government by the people”.
Direct democracy is also more representative because if the people have more power, it is obvious their interests are better represented.
Furthermore; once politicians realise the people decide, political parties quickly get together and co-operate to pass only laws and policies the vast majority of people support.
That is why in Switzerland, all major parties, of left, right and center, representing 80% of the voters, govern in coalition, always.
Of course, in Swiss elections you do not see any of demagoguery, and demagogues, we have in the US and other “representative” “democracies”.
The political, cultural, media and economic elites in representative democracies know direct democracy is more democratic and more representative, but do not want it because in a direct democracy, they will have less power, less power to make money or to reach positions of prestige.
Over the long term, those elites are politically stupid because, as the awareness of people that they are being had in representative democracy grows, unrest grows. Unrest means political instability and with political instability, the elites will lose their status, they may even lose their lives.
Black Lives Matter, Trump, the Yellow Vests, Alliance for Germany, Sanders, etc., are signs that millions on the “left” and the “right” are getting angry in representative democracy.
History shows the elites are always caught by surprise when the violent uprising arrives at their doorsteps.
Direct democracy would bring political stability to the US and other countries.
You would think the elites have lost their heads in popular uprisings often enough that they would embrace direct democracy, but they do not because their short term greed and ambition blinds them. Besides, many of them do not care if the country goes belly up; too many in the elite seem to believe countries and cultures do not matter any more; they will be surprised, once again.
If you want civic peace, prosperity and stability, push for direct democracy, like the Swiss people did many decades ago, and now have the best country in the World.
Do not take my word for it, learn about direct democracy yourself; I believe you will demand it.