confusion with add_filter

Your syntax is wrong, you’re mixing up the html and php function at the beginning.

See Filter here – wp-includes/general-template.php#L151

I think we should return the form, instead of printing it, See my example, It should work, If you want to modify something, do it in function.

Example –

function savior_search(){

    //Modify This
    $form = '<form role="search" method="get" id="searchform" action="' . esc_url( home_url( "https://wordpress.stackexchange.com/" ) ) . '" >
                <div><label class="screen-reader-text" for="s">' . __('Search for:') . '</label>
                    <input type="text" value="' . get_search_query() . '" name="s" id="s" />
                    <input type="submit" id="searchsubmit" value="'. esc_attr__('Search') .'" />
                </div>
            </form>'; 

    return $form;
}

add_filter( 'get_search_form', 'savior_search');