Having dealt with the AOK on behalf of a neighbor to get a wheel chair since he cannot walk, I couldn't tell them apart from the welfare office with regards to processes and motivation of employees. Contrasting that with a private insurance (or a housing cooperative, as you mentioned them) is day and night.
In the end, they are dependent on federal policies and subsidies, so yes, they are technically not part of the state, but for all intents and purposes, they are: funding is decided on by the state, the covered procedures are decided on by a state-created and -controlled almost-but-not-quite-state-office (Gemeinsamer Bundesausschuß, which was created by the Ministry of Health but is nominally independent-ish, yet still needs to have its decisions reviewed by the ministry, and is able to set standards and guidelines which have the characters of a law). And like all bureaucracies, they are steadily growing and require frequent budget-increases.