Here's the fix: External. Plug in your MBP/MBA SSD via External Adapter, and install Big Sur from a USB Bootable.
I had a client with a Early 2015 MBA and her issue was the same. She went to the apple store prior, and they installed a new SSD temporarily and got Big Sur to work, so they told her she needed a new SSD, because the Genius Bar wasn't able to get Big Sur to work on her current 500GB SSD, only Catalina. I ran into the same issues no matter what I tried. The issue is the internal SSD becomes Unmounted during the installation process, most likely from the installer package, and doesn't Remount.
The solution that worked for me was to plug her SSD into an external adapter, and then load Big Sur that way. I also tried a 2012 MBA SSD via External Adapter on her 2015 MBA and Big Sur installed perfect as well.
Seems like when it's plugged in externally it's not having that dismount problem.