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There's actually some cool filters that allow you to pre-populate the title field and editor field, the one we need is default_title. Once in the hook, we need to get our user and display the name:

function post_author_title( $post_title, $post ) {
    if( $post->post_type == 'your_post_type' ) {
        $user = wp_get_current_user();
    
        $post_title = $user->display_name;
    }
    
    return $post_title;
}
add_filter( 'default_title', 'post_author_title', 10, 2 );

You can try the above but if the title box is removed I'm not sure if this will work. Another option is to manually add the title once the post is saved:

Answer Link

Author - chrisguitarguy

function wpse67262_change_title( $data ) {
    if( 'your_post_type' != $data['post_type'] )
        return $data;
    
    $user = wp_get_current_user();
    $data['post_title'] = $user->display_name;

    return $data;
}
add_filter( 'wp_insert_post_data', 'wpse67262_change_title' );

Hopefully one of those functions work for you!