I definitely understand how you could interpret the two as being antithetical, but I think you're conflating the system of government with the economic system. Those are two separate things, and there are a tremendous number of combinations that can exist between the two.
In fact, democratic capitalism is one of the dominant political-economic arrangements in the world. And some of the countries that combine capitalism with the most extensive welfare states are the Nordic social democracies. They maintain market-based, capitalist economies while using taxation, regulation, labor protections, and extensive social programs to constrain some of capitalism's more destructive effects. The OECD explicitly describes the Nordic model as demonstrating that a strong welfare state can coexist with strong economic growth.
That's where I think democracy becomes particularly important. In principle, democratic institutions should give the public the ability to regulate and rein in the excesses of capitalism rather than allowing economic power to become so concentrated that it begins to dominate the political system.
And that's ultimately the concern I have with the United States: not capitalism itself, but what happens when concentrated economic power becomes sufficiently influential that it begins to undermine the democratic mechanisms that are supposed to regulate it.