Thank you @John5499, @Charday0, @Ryan2352, and @Inna22. first for naming clearly a few software names you have experience with.
I will try in the next. It is always a great help to see a review by a host.
So from my side, I tried last week with Hosthub and PriceLabs with one listing. Many things went wrong like the advanced notice (2 days) was gone. To change those, I had to jump between Hosthub and PriceLabs. The monthly calendar view on booking.com was gone as soon as it was connected to the Hosthub, etc. When I stopped the trial and canceled, more serious problems popped up. Hosthub disconnected my listings from their software, but their program didn't run properly I strongly guess. Even though the Airbnb team is working on it already more than two days, no one can book on my Airbnb listing and I as the owner do not have any access to my calendar. Guests get error messages and I cannot do anything to accept bookings. Turning on and off the instant booking work neither because the control is gone to the Hosthub.
Hosthub says they did their work correctly and I should ask Airbnb. Airbnb says Hosthub is still connected.
So what I learned so far. I think a host who is trying to find a channel manager (not too expensive) to avoid double-booking like me, should try first with a small listing which is not very important. (just make a small one) and test whether the channel manager software can connect properly and do what you want. And also disconnect and see whether they can properly disconnect leaving you all your rights back. And never test in a high season. Test it during November for example.
I was not very clever 🙂 Now I feel like I gave my car and key to a professional for a revision. They brought my car back but without the key.
Thank you again to the community for the information with names. I hope my test result provides some help to other hosts.
Best and good luck.