Social democracy isn't really socialist, it's just capitalism with some social concerns. Socialism is at its core the social ownership of the means of production; that's pretty incompatible with the private ownership of the means of production, which underpins capitalism.
This is a rewriting of history. Social democracy used to be really big on nationalizing whole industry sectors, to the point of being little different even from self-described "democratic socialists". This changed starting from the 1980s as it became clear that nationalized enterprises are almost always severely dysfunctional.
Today, social democracy mostly agrees with social liberals and neoliberals that nationalization should be used very rarely and sparingly - mostly wrt. inherently-monopolistic "platform"-like businesses such as railways, power grids or telco infrastructure. And even then, these "platforms" should be open for multiple private competitors to operate on, as opposed to providing monopolistic services of their own via vertical integration.
This is really oversimplifying things, in my opinion.
“Some social concerns” is really brushing over a lot of the things that make social democracy a form of socialism.
You’re missing for example that in a social democracy it’s common for private companies affiliated with what you could call “public service” to be in part owned by the state.
These companies most often used to be wholly state owned, but through the market liberal agenda many things have been sold out.
Sometimes for better, sometimes for worse.
A lot of what is considered welfare is also not necessarily privatized.
