This is not true. Private insurance means better care in hospitals in Germany. Better access to appointments, choice of head/experienced doctors, even can mean better rooms.
You know, I'm a med student, so I have a bit more experience :)
The Head/Experienced Doctor is still letting the Learning Doctor doing everything and pops in only for the visit.
Also the "standard" Patients get also checked by a Oberarzt regularly.
The difference is miniscule.
The Rooms are sometimes nicer I agree, but the care you get is still 100% the same. The Hospital also doesn't get extra money for the treatment besides the Fee for the "Chefarzt" and the nicer room.
Otherwise they'll only get the same DRG payout as for a "public patient"
No, it means you may potentially have all of those but there is no guarantee. Neither is there a guarantee that a publicly insured person wouldn't receive the same treatment.
E.g. if there aren't any free "better/worse" rooms what are they supposed to do? Many of these are nowadays covered by employers as a benefit or for cheap (~5€) out of pocket if you want.
Private insurance matters most for specialists that don't (aren't allowed to) have (or want) a public insurance license.