In russian wiki there is strong difference between "social state" and "socialistic state".
Second includes elimination of private property as important thing (usually on the way to communism). First is about wide social programs and state-controlled private business.
So, it may be wrong to name EU/SA state "socialistic", but they are definitely not "pure capitalism/market".
Even if we talk about "private property/business" we must remember:
- dotations and tax regulations for business, which are indirect form of STATE-CONTROLLED ECONOMY (green energy as the modern example, agriculture and farming is long-term example)
- dotations/pensions/tax regulations for private persons, which are indirect form of state-controlled "social equality"
and so on.
Also, if you remember that government is "chosen by people and express will of people" you end up with "it's people who control economy and many aspects of life in global way".
All things are mixed up in modern states.