I just tried LinkedIn on 1.20.110 and I can log in without any issues.
Few things you can try:
Clear cache completely (not just cookies)
Disable all extensions, see if it works then. Re-enable one at a time to see which one causes issues
Create a fresh profile and try to log in from that
Option 3 is the easiest and quickest to do. If that works you know there is something local that’s stopping you logging in, if that doesn’t work there is something wrong with Brave.