Did you have Windows 10 as an upgrade before you did your clean install of Windows 10? or was there 7 or 8/8.1 on there before?
If you had 10 before the clean install maybe you had it from an upgrade? would explain why it worked as Microsoft may have preserved the use of some of those older drivers or applied a working generic driver. In this case you want to consider going back to 8 or 7 (which ever was before your Windows 10 upgrade) then maybe run the Windows 10 installation again as upgrade of existing system instead of clean. If you end up doing it this way make sure your pre-Windows 10 system (7/8) has all its drivers/updates done before doing it.

*The second option in the image above is what I was referencing*
Also if you really wanted to stick with what you got there is a chance you can hunt down the media keys driver issue in Device Manager:
Try to see if you can flag it somewhere in:
- System Devices
- Sound, Video and Game controllers
- Audio inputs and outputs
It won't be the sound card you're looking for, it will look like maybe a generic name of some sort for "Media or sound" related. I can't be too sure on this to be honest but basically if you find a device in device manager related to sound or media controls that has like (!) on it or anything like "Generic" you can try:
- Right click on the device
- Properties
- Details
- Drop down: Select "Hardware IDs" (Example of the IDs "MMDEVAPI\AudioEndpoints" OR "HDAUDIO\FUNC_01&VEN_10EC&DEV_0892&SUBSYS_10438436&REV_1003")
- Copy the first result (You can right click on a ID then "Copy")
With the hardware ID you can drop it into google to do research to find out #1 WHO makes the device #2 what is the MODEL... Then make sure you take that information and enter it into the valid support site for that device.
Bad example: "Driverdownloads.com" or "Driversupport.com"
Valid site for a device by make: "http://www.realtek.com.tw/downloads/" or "http://support.toshiba.com/drivers"
Bottom line if you can pinpoint the media control driver and find the root of the making company you may have a resource from them to download a version of the driver that may work with Windows 10... I've helped clients in the past with a similar task that works but very rarely it will as some devices just stop supporting new OS drivers to a certain point.