When you say “shortcut” you mean creating an icon shortcut right? Not using a keyboard shortcut or script to open your shortcut icon? If the shortcut isn’t working, I’m guessing the path is incorrect?
It’s either the path or it actually isn’t the default browser in Windows. Or are you talking about a shortcut key combo set in the shortcut properties?
This is the path to brave.exe
C:\Program Files (x86)\BraveSoftware\Brave-Browser\Application
Have a look in your windows settings to see if the defaults aren’t overridden.
Press WIndowskey + R to open a run dialog.
Copy paste this ms-settings:defaultappsinto the box and press enter and scroll down to see if Brave is the default browser.
Also, in the default apps settings, scroll down to Set defaults by app and click that.
Then select Brave on the left side pane then click Choose defaults for this program and under protocols, is HTTP and HTTPS set to brave?
When you create a shortcut and put a URL into the box and save it, does the icon change to a Brave icon?