On versions of Windows prior to Windows 10, I can get the default browser from the following registry key:
HKEY_CURRENT_USER\\SOFTWARE\\Clients\\StartMenuInterne
Found a solution if none of these others are working. I had an issue where window's default browser directory for chromium (portable version at chromium.woolyss.com) was at the downloads folder and windows had not detected the missing executable for the default browser, icon was missing too in w10 settings.
After trying many things I eventually got a fix, updating the directory values at
Computer\HKEY_CLASSES_ROOT\Chromium(randomstring)\shell\open\command
at "\HKEY_CLASSES_ROOT\Chromium(randomstring)\" the random letters/numbers at the end of chromium will be different for everyone im assuming so just look for chromium and you'll see it.