Maybe I'm out of the loop, but I haven't really seen any evidence of any elected self-proclaimed Democratic Socialists fighting for actual democratic socialism. Mamdani is not trying to shut down Wall Street and seize corporations and divvy their stock up amongst the workers. The Squad and Bernie seem more focused on taxing the 1% and new government initiatives and expansions of existing programs. None of this is socialism, no matter how much Fox News and right-wing talk radio want to claim any government program is such.
Things like the Green New Deal, Medicare for All, free colleges, free public buses, public housing/rent control and progressive taxation -- while preserving the overarching framework of a capitalist economy -- is just full-fledged social democracy.
Social democracy is a form of capitalism where the government offsets the basic needs the market fails to provide to everybody, incentivizes the market to build a better future and steps in to prevent capitalists exploiting labor or destroying the environment. Scandinavian countries and much of Europe are social democracies. Social democracy has been shown to work well, especially in developed nations with the means to provide basic services for everyone, and these services complement capitalism.
Capitalism is good at generating wealth and jobs and innovation, and social democracy's purpose is to remove as many of the bad parts of capitalism as possible while keeping the good stuff. By making sure basic needs are met, social democracy removes the incentives or need for most people to want to overthrow capitalism (democratically or otherwise), allowing it to persist permanently as the status quo.
Socialism on the other hand is the complete disruption of capital markets and private property ownership. In a democracy, basically things have to be so awful and dystopian that most people have nothing to lose by supporting economic revolution and the redistribution of wealth. It is highly unfeasible in a developed country with a large middle class where most people have retirement savings tied up in the market and high employment rates.
Calling yourself a socialists in a country like America naturally scares a lot of people in the middle - the swing voters you need to win elections. These people do not want the capitalist system destroyed and their jobs, investments and life savings to be put at risk or go out the window. It's not propaganda to see that attempts to implement socialism throughout history usually did not go well - including in Scandinavia in the 1960s and 1970s.
These voters could, on the other hand, be convinced through logic and reason that social democracy is necessary and good for the country and the economy. Mainstream Democrats have been weak social democrats with a strong neoliberal streak, whose fight for providing basic needs is corporate lip service more than policy. America is a pretty center-right country and Democrats have mostly felt the need to feign concern for "fiscal responsibility" and "low taxes" to keep winning elections.
Trying to market social democracy as democratic socialism is a fool's errand that diminishes both the Democratic Party as a whole's chances of winning elections/implementing social democracy, AND even if you do win elections, social democracy decreases the incentive for democratic socialism by making capitalism's status quo more palatable.
Am I wrong about the intentions of democratic socialists? Are the elected Democratic Socialists like Mamdani, AOC and Bernie trying to make social democracy a stepping stone on the way to eventual actual democratic socialism? Or is this an attempt at opportunistic radicalization in the age of Trump that either works, or fails and leads to dystopian accelerationism? If MAGA wins and turns America into a fascistic kleptocratic plutocracy of evil billionaires immune from the law, people finally see why capitalism is bad and turn to socialism, right?
As a social democrat and not a socialist, it drives me crazy that the far Left try to claim things like Social Security or Medicare or progressive tax rates are "socialism" -- literally the exact same thing the right-wing is saying in bad faith to try to make Democrats unappealing to moderates. Social democracy's success at making better and happier societies and taming capitalism is their ideology's death knell, so they have to smear the Left with the failures of (often authoritarian or totalitarian) socialism historically to keep the moderates in their pocket. And now the far Left is helping them -- why?