About 95% of the money spent by the public health insurance company is „Leistungsausgaben“, I.e., paying out people for health related costs.

You can’t optimize that away, even when combining the companies. The remaining 5% is overhead. Having worked in a big company, I can tell you that big companies are not that much more efficient than small companies. In fact, the overhead is often even larger since there is lots more cross-communication involved between departments. In the end, everyone that is now a CEO would be an SVP instead, and barely anything would change.