The problem is that we're trying to achieve a goal which is not possible with the tools at hand. There is no way to satisfy the right of all people to self-govern in a nation where half of the population wants a secular society with a European-style welfare system, and the other half wants a laissez-faire system that imposes Christian social mores. The goals of the left and right are not reconcilable.
The underlying issue here is that democracy is not enough. To state it differently, democracy is a means to an end, with the 'end' being self-representation. Democracy is necessary to achieve self-representation, but it's not sufficient.
An example: remember when The Don wanted to buy Greenland? Well, let's say the US bought Greenland. Their people, then, would be granted US Citizenship, including the right to vote. Throughout the transition, they would reside in democracies. One day they would stop voting in Danish elections and would instead vote in US elections. They would always be able to participate in the democratic process. They would also be extremely pissed off. Why, though? They had their say, didn't they? They voted in the elections, didn't they? But even though the process was democratic, the right of the people of Greenland to self-govern was not observed. They had democracy, but not self-representation.
Folks in the US feel the same way. In the Pacific Northwest, there is a broad consensus that the state should run healthcare, and yet they don't have state-run healthcare. Why? Because they need the consent of folks who live thousands of miles away in Oklahoma, Texas, and Florida! They have democracy, but they don't have self-representation.
When the tide turns and there is majority support, on the national level, for state-run healthcare, what happens in Oklahoma? We're going to force a national healthcare system on them when the vast majority of the citizenry abhor the idea of state healthcare. And why? Because, thousands of miles away in California, Oregon, and Washington, folks support the idea. That's democracy, sure, but it's not self-representation.
The rash of democracies that have taken hold in the world over the last few centuries has been wonderful, but we'd be foolish to assume that we've perfected human governance. There are better, more representative forms of government out there that we need to seek out and implement.
To make the next great leap in human governance, we need to acknowledge the limits of the systems that we currently have. We need to admit that over-large democracies are not conducive to actual self-representation. And we need to exorcise the ghosts of our monarchical forebears. Our governments don't own us, do they? Then why do we allow them to stifle and suppress democratic secession movements, like the recent one in Catalonia?
To ensure that we are truly able to self-govern, we need to pick our governments, not just our representatives.