A quick fix I did was plug the headphones that came with my phone into my computer and set them as a default microphone while using my headset for the output. Worked flawlessly
I gave you a heart 
I will give a heart for every bluetooth issue like this. …I think they will only care about this if your bug report gets a hundred or a thousand hearts 
this problem still happens in 2022! amazing!
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August 31, 2023 - Our requests to the intern at Blizzard HQ continue to go unheard. We are losing hope that help will ever arrive. Tell my headphones that I love them, and that the aux cord broke protecting their future. Good bye.
Works like a charm, thanks!
December 2023 - This is still happening. I set input device to None in the Blizzard launcher settings, but SC2 still takes control of my mic and enables handsfree mode, making the audio quality absolute garbage. I am amazed how this goes completely ignored. It literally ruins my Starcraft experience and it’s not possible to fix it anymore since the voice chat options were removed.
Can this PLEASE be adressed? I’ve purchased SC2 and the extensions when the game still cost money, but I’m not sure if I want to get Blizzard games in the future when such basic and impactful problems are not adressed.
Every other game works fine for me, from 20 year old classics to modern titles. Only SC2 has this problem and I am very disappointed that Blizzard is not even responding.
You can still disable the headset device in windows (either completely or just for all desktop applications). It’s annoying because then you have to enable it back if you want to go to voice in discord or any other app, but possible.