TL;DR: At least for me, the only solution is to downgrade to Windows 10 version 1709
Update: Welp, that didn't work either...
It's a fact that you can't use a Bluetooth headset's mic and stereo audio at the same time, because stereo needs two channels and mic needs one. You can't use more than two channels AT ALL with Bluetooth technology. That's why with apps like Skype, the Hands-Free Profile
needs to be used for both mic and audio.
But I have the same problem with my Samsung U Flex headphone even if I use the Hands-Free Profile. I have tried every solution on this thread and everywhere else and nothing works.
I also found out that some headsets get muted automatically when idle and you have to open Sound Mixer and mute and unmute your headset when in a Skype call or press a combination of keys on your headset to effectively unmute it. This sounded about right
because the sound bars of my HF audio in Sound Control Panel would rise up and down but I just couldn't hear anything. However this still didn't work for me and my headset doesn't have a combination of keys to make it mute/unmute.
To my surprise, the same headphone works flawlessly on my friend's laptop which has Windows 10 version 1709. I took a look at the headphone drivers in his Devices and Printers section which looked something like this:

Which looks totally different from mine (I have Windows 10 version 1903). At first I tried to force install the same drivers that he had, but came across some drivers that can't be installed for versions of Windows 10 higher than 1709 (like e.g.
Bluetooth Audio/Video Remote Control HID)
Seems like Microsoft has messed up a working functionality as usual. I guess I have no choice but to fresh install a 1709 version of Windows 10 until this gets fixed. What a pain.
Update: I just installed a 1709 Windows 10 and the same problem persists. I guess it's just my laptop's Bluetooth chip. I think I'll buy a Broadcom Bluetooth adapter or something.