There are two kinds of private schools: staatlich anerkannt: equivalent to a state/city school they follow the state curriculum. Pupils can change whenever they like to a state school, their grade reports are accepted and they do the same graduation exams (Mittlere Reife if they end school at 16, Abitur which allows them to go to university at age 18) as the pupils at state schools. They can change without any extra exams to state schools if they want to. staatlich genehmigt: these schools are not allowed to hold graduation exams, their pupils will have to do that exam at a state school and it is doubtful if they will pass it. Bavaria for example has a Zentralabitur, this means that the state centrally sets down the exam questions each year in Bavaria and that all pupils in Bavaria write the same exam that day. The Abitur exam in Bavaria is hard to pass, only 20% of all children in Bavaria go to the Gymnasium and will pass the Abitur.
A child who has not been learning according to the state curriculum has almost no chance of learning what he/she needs to pass the Abitur on their own except if he/she is very intelligent and then there would have been no reason not to have that child attend the state school in the first place. Because that's the reason German children attend the "staatlich genehmigte" schools, they can't make the cut in the normal schools. If a pupil at a "genehmigte" school wants to change to a state school or an "anerkannte" school, they have to pass an admission exam because their prior schooling isn't recognised.
As you said, for expats only staying a few years these schools are a good alternative because their children will not be stressed by an education according to the strict German system and since nobody can ever fail a year there, there will be no need for private tutoring.
source Bayrischer Rundfunk:
Grundschule (=primary school, age 6 to 9) official lists for Munich: state/city (Staatliche/Städtische) Grundschulen staatlich anerkannte Grundschulen staatlich genehmigte Grundschulen
Gymnasium (= high school, age 10 to 18), official list of
state/city (Staatliche/Städtische) Gymnasien staatlich anerkannte Gymnasien staatlich genehmigte Gymnasien in Munich, see right-hand column of this page