On the Mac you need a mic (or a headset) with "power" (e.g. a USB mic or a pre-amp mic).
Conclusion: I have stored away my headset and will use the built-in speaker/built-in mic to make calls. In case I need privacy I will just plug in my iPhone earbuds. That sounds like a/the winning configuration. But that also means that I do not have a solution for environments with a lot of background noise. Maybe I need to go back and need to find out why the quality with the JawBone was bad in the first place.