Earlier quoted context omitted.
I think that's true. But it's also that he Swedish concept of welfare has traditionally included the broader concept of "the welfare state". Meaning that welfare doesn't just apply to social security as such but the availability and quality of housing, education, care, transport and, as you say, even recreational activities. It's not that sport organizations in Sweden are open to everyone out of the goodness of their…
Regarding sports there is nothing stopping anyone from starting an entirely profit driven club, like most gyms for example, as far as I know. I think it's mostly a matter of tradition. And the financial support you get for that type of broadly run organisation, not just sport. But that too speaks to the egalitarian traditions.
For instance the school system isn't very egalitarian, at least not in Stockholm, these days (nor is the housing market for that matter). The school system is quite different compared to e.g. Finland. The laws changed and so did the quality of different school. "Grade inflation" is very common etc.