We are also in Germany (down south). The first year or so was difficult but got progressively easier. The best thing about local schools here, IMO, is that when they are ten, they change to the Gymnasium/Hochschule/Mittelschule. In the east, where we are going to move to, the school is all in one, so the kids can, theoretically, move to which suits them best.
We were recommended by another expat family to not go to the international school because their own children had real problems integrating and their German still wasn't perfect after several years. This was because they spoke predominantly English in the school. So we ended up enrolling ours in local schools.
Overall, in terms of integration, I think we made the right decision to go local. Our kids do attend different schools. One is overwhelmingly better than the other but my husband says if they have a good teacher, and they are learning, then that is the most important thing. Plus, the local schools offers great extracurricular activities from the Grundschule up, like debating and school newsletter writing (great for improving German).
There are also the local Sport Verein which have things like handball, soccer, gymnastics, judo and so on, which are great places to build friendships, as well as the girl guides/scouts clubs (called Pfadfinder).
Small tip too. Parents often want their German kids to make friends with native English speakers, because they want their kids to be as fluent as early as possible.
The other things that probably has made life more difficult is this corona situation here. We have spent so much time home-schooling, including recently, and it has been difficult supporting the kids in another language, regardless of our own language and education level.
It is often hard work, isn't it, but I hope it gets better. That said, it isn't for everyone. I wish you all the best in figuring this out.