Access to specialists mostly need a referral from your family doctor.

That hasn’t been the case in many years now, you can just make an appointment without a referral.

Well, for most specialist. There are still a few cases were it is required, e.g. radiologist.

Also with public insurance you might have to wait longer for your appointment with a specialist - but if you have a referral from your family doctor, they can add a urgency note (Dringlichkeitsvermerk) on it and you will get an appointment faster. With that you can also call the health service hotline (116117) and they have to find an appointment for you.