The Nordic countries implement "social democracy" in some form or other. At least they used to, as you mention Sweden has seen its economy liberalized far beyond the US in some sectors these days (Schools, for example).

In the end it's just a question of whether Social Democracy with a free market is half socialism or if it isn't. Not really a useful debate. The roots of social democracy just like all other forms of socialism of course is the workers movement and Marx is very present still in the Social democratic parties' program declarations.

Social democrats accept the market economy. They are socialist in the sense that they are proponents of redistribution/wealth equality, and workers influencing or even owning the means of production (The former through unions having seats in corporate boards, the latter through later scrapped ideas like workers owning corporations through funds, "wage earner funds").

These social democratic ideas try to adapt Marxist theory to a liberal democracy and market economy. Is social democracy socialism despite accepting a market economy? Depends on who you ask. It's simpler to just use the term social democratic instead of socialist. Having a large welfare state is really orthogonal to how the money for said welfare state is raised (Norway for example could in theory have supported its welfare state without much social democracy or socialism).