Unable to install version 3.0 in Drupal 10, but the site says it is compatible for Drupal 10.

Huh. I'm not quite sure why it is trying to get 3.0.0 -- it is 3.0.1 that adds D10 support.

Could you post the output of composer why-not 'drupal/social_auth_facebook:^3.0'?

I am getting the below output while running

composer why-not 'drupal/social_auth_facebook:^3.0'

I have tested the 3.x version in D9 successfully. From the error message it sounds like you may be missing some dependencies of a proper Composer setup for Drupal. I recommend taking a look at Using Composer to Install Drupal and Manage Dependencies → .