using wp_set_auth_cookie with out any hook

Wordpress version 4.7.5

I am signing up new users from Instagram, I am successfully able signup the new user. after signup I am trying to logged in the user.

I can use hook, but would like better solution than the dirty one, I have posted as that will be totally unuseful .

if( $user_created ) {
    $new_user = get_user_by( 'id', $user_created ); 
    //$loggedin = programmatic_login( $new_user->user_login );
    //ob_start();
    if ( !is_user_logged_in() ) {
        wp_clear_auth_cookie();
        wp_set_current_user( $user_created, $new_user->user_login );
        wp_set_auth_cookie( $user_created, true );
        do_action( 'wp_login', $new_user->user_login );
        // wp_safe_redirect(site_url() . '/profile/');
        // exit;
        // $redirect_to=user_admin_url();
        // wp_safe_redirect($redirect_to);
        // exit();
    }
    //ob_end_clean();
} ?>
<script>jQuery(document).ready(function(){ window.location.href = "<?php echo site_url() . '/profile/'; ?>";});</script> <?php

also tried with a custom programmatic_login function from another post.

if I am trying to var_dump wp_get_current_user() and $_COOKIE below this, I am getting user object for wp_get_current_user() and array(1) { ["wordpress_test_cookie"]=> string(15) "WP Cookie check" } for $_COOKIE .

Important Edit

The code is inside a function, which is hooked to a hook, that is called in side a page that is after header is getting displayed, so we could not use init here. any other way of doing it and also because of this wp_safe_redirect() or header('Location: ') also not working .

Important Update with what I have just tried

A dirty work around

when I created the user from Instagram login, After successful creation I redirected the user to a page with get parameter something like ?user_id=$inserted_id and on wp hook, I tried to get the GET['user_id'] and logged it and it worked, trying to find a proper solution.

if ( $wpdb->insert_id ){  //that is registration with instagram
   ?>
    <script>jQuery(document).ready(function(){ window.location.href = "<?php echo get_bloginfo('url') . '/profile/?id=' . $wpdb->insert_id; ?>";});</script>
   <?php
}

and then

add_action( 'wp', 'login_current_user' );
function login_current_user(){
    if ( is_page() && get_the_id() == 863 ){
        if ( isset( $_GET['id'] ) ){
            if ( !is_user_logged_in() ) {
                $user_id = $_GET['id'];
                $user = get_user_by( 'id', $user_id ); 
                wp_clear_auth_cookie();
                wp_set_current_user( $user_id, $user->user_login );
                wp_set_auth_cookie( $user_id, true );
                do_action( 'wp_login', $user->user_login );
                if ( is_user_logged_in() ){
                    $redirect_to=site_url() . '/profile';
                    wp_safe_redirect($redirect_to);
                    exit();
                }
            }
        }
    }
}

Proble with dirty trick. If the id we are passing as get parameter exists in wp_users table then with no verification that will be logged in .

Update

I created a user_meta before redirecting to profile page and after verification login the user and removed the usermeta, just like an OTP . Trying to make it better if possible.

Below the code:-

if ( $wpdb->insert_id ){  //that is registration with instagram
    $temporary_token = sha1(rand());
    update_user_meta( $wpdb->insert_id, 'temporary_token', $temporary_token);
   ?>
    <script>jQuery(document).ready(function(){ window.location.href = "<?php echo get_bloginfo('url') . '/profile/?id=' . $wpdb->insert_id . '&token=' . $temporary_token; ?>";});</script>
   <?php
}

and then

add_action( 'wp', 'login_current_user' );
    function login_current_user(){
        if ( is_page() && get_the_id() == 863 ){
            if ( isset( $_GET['id'] ) ){
                if ( !is_user_logged_in() ) {
                    $user_id = $_GET['id'];
                    $user = get_user_by( 'id', $user_id );
                    if ( $_GET['token'] == get_user_meta( $user_id, 'temporary_token', true ) ){
                        delete_user_meta( $user_id, 'temporary_token', $_GET['token'] ); 
                        wp_clear_auth_cookie();
                        wp_set_current_user( $user_id, $user->user_login );
                        wp_set_auth_cookie( $user_id, true );
                        do_action( 'wp_login', $user->user_login );
                        if ( is_user_logged_in() ){
                            $redirect_to=site_url() . '/profile';
                            wp_safe_redirect($redirect_to);
                            exit();
                        }
                    }
                }
            }
        }
    }