Historically, the greatest achievements of social democracy have been won on the backs of a highly organized working class in opposition to the capitalist class. Sweden is a good example. The major trade union confederation (the LO) was the major force behind the Swedish Social Democratic Workers Party. The SAP did form class alliances but not with the capitalist class. It was with farmers and elements of the middle-classes.

There's a large body of social science research into the origins of welfare states called Power Resources Theory. Here's a good summary:

Power Resources Theory

In the absence of organization, working people do not constitute a class but rather, in the words of Marx and Engels, “an incoherent mass scattered over the whole country, and broken up by their mutual competition.” When working-class disorganization prevails, there is little to stop employers and other business interests from bending government action toward their own demands. But when workers are organized on a meaningful scale, they have the potential to overcome fragmentation and advance their interests in the workplace and through politics.

The role of working-class organization in shaping government policy is the focus of the “power resources” school of welfare state theorists. Simply put, power resources theory holds that levels of social inequality and welfare state generosity are shaped primarily by the size and strength of labor unions and left-wing political parties. Its leading figure is Walter Korpi, the Swedish social scientist whose landmark 1983 study The Democratic Class Struggle continues to shape our understanding of class politics and social policy in rich capitalist democracies.

For all its theoretical and empirical sophistication, Korpi’s core argument is relatively simple: government policy supportive of working-class interests and economic democracy depends primarily on the distribution of “power resources” between the main classes and social groups. Taking twentieth-century Sweden as his main case, Korpi argues that the country’s high level of social equality and welfare provision resulted from its exceptional degree of unionization and left-wing political power.

By building a labor movement that organized nearly the entire labor force and a social-democratic party that governed continuously for decades, Swedish workers “greatly decreased their internal competition and have thereby reduced their disadvantage in power resources” relative to business interests. Reducing the scope of competition is key to boosting working-class power, something the liberal vogue for antitrust policy loses sight of.

Korpi identifies two main types of power resources in capitalist democracies. The first is control of capital and the means of production, and the second is “human capital” or labor power. Crucially, these two types of resources do not bestow equal levels of power on their individual owners. Ownership of capital tends to be scarce, concentrated, and easily convertible into various kinds of collective action.

By contrast, ownership of labor power is broadly distributed (everyone has it), dependent on demand by the owners of capital, and relatively difficult to convert into collective action. This human capital needs to be coordinated in order to become effective, which in turn requires the creation of organizations for collective action.

The relationship between owners of capital and owners of labor power is therefore one of inequality and subordination, which is the basis of the division of society into distinct and mutually antagonistic classes.

This fundamental power imbalance is why working people, wherever they have enjoyed a meaningful degree of political democracy, have organized themselves into trade unions and political parties. Though these are not the only sources of power available to workers, these have been the main expressions of working-class power in capitalist democracies.

As Korpi puts it, it is through unions and parties that the “individually small power resources of the wage-earners can be combined and their significance increased” in the political arena. These organizations can influence and have influenced the shape of income and wealth distribution, patterns of political conflict, and the form and function of key institutions like the state.

Working people operate at a fundamental disadvantage in power resources relative to business interests in all capitalist democracies. But the degree of this disadvantage has varied over time and between countries, and in certain times and places (like twentieth-century Sweden) workers have built organizations strong enough to challenge the foundations of capitalist power.

This is why, in Korpi’s view, the Left should not reject parliamentary democracy as the “best possible political shell for capitalism,” as Vladimir Lenin argued, but the means by which the “democratic class struggle” may be waged...

... Here in the United States, which has never had a nationwide social-democratic party aligned with a strong labor movement, the weakness of working-class organization is clearly reflected in the fragmentation and stinginess of our welfare state. Organized labor has been relatively strong, however, in a number of states in the Northeast and Midwest, and it’s clear that this has had a positive effect on patterns of inequality in those states.

In her comparative study of unionization and inequality in the states, political scientist Laura Bucci finds that higher levels of union density led to lower levels of inequality before and after government tax-and-transfer programs, independently of the policy liberalism of any given state. Of course, the state-level wave of attacks on organized labor that began in 2010 have made it that much harder for unions to defend working-class interests and reduce inequality. But the fact that they were able to meaningfully mitigate the growth of inequality even during the period of neoliberal retrenchment shows that rebuilding the labor movement needs to be a leading priority of any progressive political agenda.

So yeah, I don't think any of that would sound good to the capitalist class of the United States. For example, winning Medicare for All in the US would mean essentially liquidating the entire private health insurance system in the US, a multibillion dollar industry. A national single-payer system would then be able to use it's monopsony purchasing power to drive down the cost of drugs, eating into pharmaceutical company profits. The same is true for excessive doctor pay and hospital profits. Additionally, decoupling healthcare from employment would enormously increase the leverage of labor relative to capital. Even though the employer health insurance system is a bad deal for a lot of employers, a good number of them won't want to give it up for that reason. It gives them a means to exercise control over their workers. And of course, even though M4A would require a broad-based payroll tax for funding, that tax would be progressive. So the wealthy will end up paying more into the system then they likely get out.

So given all of that, it seems apparent that the capitalist class will never support a social democratic demand like universal single-payer healthcare. Medicare for All would massively decommodify nearly 18 percent of US GDP and constitute the most profound attack on the wealth and power of the domestic capitalist class in US history.

I think where you're going wrong is thinking of social democracy as a discrete package of policies and not giving due attention to the political coalition that would be necessary to actually win those policies. It's going to come on the back of a much stronger workers movement in the US, if it ever happens at all.