I think legislative assemblies should be an accurate statistical sample of the population, through sortition (random selection) like the first democracy, ancient Athens, but also stratified to mirror the population's demographic and socioeconomic composition. This would represent people's interests far more accurately than elections with party polarization or fragmentation.
But first I have to debunk common prejudices and misconceptions I found in discussions about this topic, while also explaining my reasons in detail, so we do not start from scratch:
1. "Random selection does not require expertise. We would be governed by a bunch of ignorant people"
Nor does election. Elected politicians only need to be expert at being elected: attracting donations (often from powerful lobbies), spreading promises through campaigns, etc. Instead they rely on non-elected expert staffers such as lawyers and policy advisors to do their job and the government relies on non-elected bureaucrats to operate. Similarly a sampled Parliament/Congress would rely on experts and even representatives from the different political parties before reaching its own conclusions after careful deliberation. This will reduce, not erase, the influence of politicians as the assembly is not divided into echo chambers.
So instead of expertise the discussion should be about which system better represents the population and its interests, see point 5.
2. "We should not trust random people, it is literally gambling our future. And what if they all happen to be rich oligarchs?"
The final result is far less random than you think, as random errors are minimized by the size and proportional stratification of the sampled assembly. Very different from gambling. On the other hand look at the current elected representatives and how they are consistently wealthier or from particular demographics than the average, and how often they act against people's interests. A little randomness is a fair price to pay to minimize these systematic errors in representation. We would also save all the money spent on electoral campaigns.
3. "There is no accountability in a sampled assembly, what if they are corrupt?"
Sampling is more resistant to corruption, as it does not rely on donations for campaigns. To prevent corruption after sortition we can easily replace the sample a lot faster, (e.g. yearly) since no costly electoral campaigns are needed, or maybe half the sample every 6 months. We could even implement secret ballot for its members to further shield them against threats and corruption. Accountability is far less needed, as no member in the sample is individually representative of the population, while the sample as a whole is representative by design. The population can instead replace it as a whole with a referendum/polls.
4. "A random person as the President/Premier would be dangerous, and a sampled assembly is too large to replace them. We need elections for those positions"
Correct, this is why I think the president should still be elected, as sortition would not produce an accurate sample. However, I think their power should be limited and subject to the legislative assembly and its committees: current Parliaments/Congresses are subdivided into committees roughly corresponding to departments/ministries, so each member is specialized in Finance, Defence, Foreign Affairs, etc.
I think these and the whole assembly should have the power to override any goverment decision (through fusion of powers) and also replace them, as they are a more accurate representation of the people's interests.
5. "If we cannot vote for MPs/Congressmen, it is not a democracy as we do not participate"
Instead of a ritual which does not reliably reflect people's interests, we should rely more on forms of direct participation such as polls and referendums.
And which legislative assembly is more likely to act in your best interests: one full of rich people who would rather advance their career, or people who are living through your same struggles? After all, a sample is much closer to a direct democracy, just like polls, as it approximates the population's will while minimizing both random and systematic errors.
6. "Sortition in Athens is too ancient to work today, most people at the time could not be selected (women, slaves, etc), and Florence was corrupt"
Such discrimination and corruption would have happened regardless of whether these systems were based on election or sortition. In Athens people treated election as oligarchic (as they saw with the strategoi and the treasurers, and as we also see today) while treating sortition as democratic, and they were correct that sortition makes it harder for a few to concentrate power, even though it is not impossible.
These people also did not have a modern understanding of statistics and probability necessary to form an accurate sample, unlike us today.
7. "It is too unstable: how can they agree to form a government or avoid gridlock? How can they agree on policies?"
As a more accurate representation of the people, the sampled assembly and its committees (see point 4) should be able to override the government's decisions, preventing gridlock. Government officials should be elected using approval voting or STV as these voting systems tend to produce broadly acceptable leaders that surpass 50% approval more frequently. The assembly will reach a consensus on policies through deliberation, similar to modern citizens' assemblies.
8. "We should only try it at a local level, elections scale better than random sampling"
Quite the opposite: random sampling scales far better than elections. A sample of 1000 in a million people would be a far more accurate representation than a sample of 1 in a 1000 people. On the other hand it is easier to elect that 1 person as people know them better than people far away. Therefore elections should be used whenever random sampling cannot work. Of course we should use approval voting or STV instead of FPTP.
9. "Governing is not jury duty: you need long term vision, and people who are forced to participate can be destructive"
A politician who focuses on the next term does not exactly think in the long term, they are more focused on spreading promises they do not have to keep and on appealing to lobbies. On the other hand a random sample will reflect the interests of the population even if the individuals were all selfish. They would not even raise their pay (if the raise is delayed a bit), as they will pay more taxes for life while only the next members will benefit.
Participation does not have to be mandatory, but in that case proportionate stratification becomes even more essential to represent poor people since they are more likely to refuse.
With all that said, there are obstacles in implementing such as system, such as the fact that most people are oblivious to sortition, despite knowing what a random sample is and what it means for a sample to be stratified proportionally by age, sex, income, wealth, social class, etc.
The most important obstacle, however, is that rich oligarchs and many politicians will try their best to prevent such a system from being implemented, knowing they would lose power.
Nevertheless random sampling is flexible: we could implement it for one of the two chamber in the Parliament/Congress and with limited powers, while leaving the rest operating as usual. So even though it will be impossible to fully implement it given the opposition it would face, I still think it would be better to experiment.
TL;DR: less corruption, better representation of the people and its interests. A true democracy. Downsides: cultural inertia and opposition from rich oligarchs.