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 hearts (though I think many people support it). It's because they are required to if they want to be registered as non-profits, receive government funding, rent subsidized facilities owned by the state etc.
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.
What's stopping them is of course that they can't compete with non-profits that are being subsidized by the state. (Which I don't disagree with). Sure, it is a tradition, but like many things it's enabled by the laws on the books.
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.
Ah, but they would be able to compete with the subsidised, volunteer run organisations if there was greater willingness to pay for a more professionally run club, with better facilities etc. This does happen for some sports, such as gyms where there is a market for upscale clubs, but not for football for example, to my knowledge. People would just feel weird about putting their children in a luxury football club.
Every year when it's time to sign the children up for ballet, football, gymnastics etc it's the same thing, extremely high pressure on all clubs, places fill up within hours. I'm sure that if it wasn't for Jante, there would be a huge market for more expensive clubs with shorter queues.
In what sense is the school system not egalitarian? It's not very good anymore, true, but that has other causes. I mean all schools cost the same, i.e. nothing, and anyone can apply to any school. It's even forbidden to charge for school lunch, or even ask children to bring fruits because "some people may not be able to afford that" (though this is going way too far in my opinion).
I agree that housing has changed, and prices are getting pretty crazy in places. But regardless of the price differentials between areas, which still are far from as extreme as in the US, my point is more that it doesn't matter as much which area you live in, your quality of life will be pretty similar. Building standards are extremely high in Sweden so even the cheapest places are pretty robustly made at least. And the houses on the most expensive addresses are still relatively modest, there are no doormen or valets or crazy things like that. The differences are more subtle, and there is less showing off.
It's not that sport organizations in Sweden are open to everyone out of the goodness of their hearts (though I think many people support it). It's because they are required to if they want to be registered as non-profits, receive government funding, rent subsidized facilities owned by the state etc.