It is not the case that GitHub has changed to allow only SSH keys for authentication. GitHub used to allow users to use a username and password over HTTPS if they were not using 2FA, a username and personal access token over HTTPS, or SSH using an SSH key. The only thing that has changed is that you can no longer use a username and password for HTTPS; you must use a personal access token instead of a password if you wish to use HTTPS.
If you give another user one of your personal access tokens or add one of their SSH keys to your account, they will have access to all of your repositories. This is insecure, and so you should not do it.
Instead, you should grant your contributor access using the Manage Access interface, and make sure they can access the repository using their own account. If they are using HTTPS, then they may need to either switch to SSH by changing the URL with git remote set-url git@github.com/owner/name.git
(replacing owner
and name
) or just follow the directions outlined in this answer.
The fact that a contributor cannot access their own account is an issue that they need to address instead of having them access your account.