Add custom checkbox in admin columns with according ajax event for saving postmeta.

column-checkbox.php

<?php class Post_Column {
	function __construct() {
		add_filter( 'manage_posts_columns', [ $this, 'columns_head' ] );
		add_action( 'manage_posts_custom_column', [ $this, 'columns_content' ], 10, 2 );
		add_action( 'admin_footer', [ $this, 'jquery_event' ] );
		add_action( 'wp_ajax_newsletter_send_save_meta', [ $this, 'process_ajax' ] );
	}

	/**
	 * Add column header
	 *
	 * @param $defaults
	 *
	 * @author Maxime CULEA
	 *
	 * @return mixed
	 */
	public function columns_head( $defaults ) {
		$defaults['newsletter_send'] = 'Newsletter (envoyée)';

		return $defaults;
	}

	/**
	 * Add the column checkbox
	 *
	 * @param $column_name
	 * @param $post_id
	 *
	 * @author Maxime CULEA
	 */
	public function columns_content( $column_name, $post_id ) {
		if ( $column_name == 'newsletter_send' ) {
			echo '<input type="checkbox" class="newsletter_send" data-postid="' . $post_id . '" ' . checked( 'yes', get_post_meta( $post_id, 'newsletter_send', true ), false ) . '/>';
		}
	}

	/**
	 * jQuery event to send the checkbox value
	 * only for post archive admin page
	 *
	 * @author Maxime CULEA
	 */
	public function jquery_event() {
		global $pagenow, $typenow;
		if( 'edit.php' !== $pagenow || 'post' !== $typenow ) {
			return;
		}

		echo "<script type='application/javascript'>jQuery(function($){
			$('.newsletter_send :checkbox').change(function(){
				var checkbox = $(this);
				$.ajax({
					type: 'POST',
					data: {
						action: 'newsletter_send_save_meta',
						value: checkbox.is(':checked') ? 'yes' : 'no',
						post_id: checkbox.attr('data-postid'),
						newsletter_send_nonce : '" . wp_create_nonce( "newsletter_send" ) . "'
					},
					url: ajaxurl,
				});
			});
		});</script>";
	}

	/**
	 * Check Ajax event to save the checkbox value as post meta
	 *
	 * @author Maxime CULEA
	 *
	 * @return bool|int
	 */
	public function process_ajax() {
		check_ajax_referer( 'newsletter_send', 'newsletter_send_nonce' );
		if ( empty( $_POST['post_id'] ) ) {
			wp_die( -1, 403 );
		}
		return update_post_meta( (int) $_POST['post_id'], 'newsletter_send', (string) $_POST['value'] );
	}
}

new Post_Column();