Ok guys, I have just fixed it.
I decided to check my headset on my wife's laptop. She didn't use it for a long time so I started to experiment. I connected my headset to it and all worked fine.
I watched the driver version, of course, it was quite old - 10.0.22621.608. I decided to try to roll back somehow my drivers to this.
I googled the version and it brought me to the https://www.intel.com/content/www/us/en/download/19725/realtek-high-definition-audio-driver-for-windows-10-64-bit-and-windows-11-for-intel-nuc-11-performance-kits-mini-pcs-nuc11pa.html
I just downloaded the Audio-Win10_Win11-6.0.9360.1_SST-6367.zip driver and launched the installation.
First, it uninstalled my current drivers and rebooted. Before it proceeded to the installation I checked my headset and the mic worked! I continued the installation, but it finished with some error popup I don't remember the message.
In Device Manager I found that Intel High Definition Audio device is not recognized and I decided to update it with the help of Windows Update tool.
After that, unfortunately, my mic didn't work again.
I ran Audio-Win10_Win11-6.0.9360.1_SST-6367.zip driver installation again, but instead of searching for the driver for Intel High Definition Audio via Windows Update, I searched it by device code (for me it - "intelaudio func_01&ven_10ec&dev_0294") and it brought me to the https://www.intel.com/content/www/us/en/download/19243/realtek-high-definition-audio-driver-for-nuc7cjy-nuc7pjy.html page where I downloaded Audio-Win10-Win11-6.0.9492.1.zip driver.
After it was installed, all my audio devices worked as they should.
So, try to do something like this, maybe it will help you.
BTW, After my wife's laptop was updated my headset continued to work fine on it. And I tried to connect another BT headset to my laptop, it also worked fine while my headset failed.