Can I add a ‘show all pages’ link to wp link pages?

Here’s a suggestion using the wp_link_pages and content_pagination filters:

The aim is to add a “Show All Content” link below the content pagination links:

and when we click it, the link should display “Show Content Pagination” and vice versa.

Demo plugin

We can do that with the following demo, where we introduce the wpse_show_all_content GET variable:

add_filter( 'wp_link_pages', function( $output, $args )
{
    // Validate user input
    $show_all_content = (bool) filter_input( 
        INPUT_GET, 
        'wpse_show_all_content', 
        FILTER_VALIDATE_BOOLEAN 
    ); 

    // Generate link
    $link = add_query_arg( 
        'wpse_show_all_content', 
        ! $show_all_content, 
        get_permalink() 
    );

    // Generate link title
    $title = $show_all_content 
        ? 'Show Content Pagination' 
        : 'Show All Content';

    // Append    
    $output .= sprintf( 
        '<a href="https://wordpress.stackexchange.com/questions/239209/%s">%s</a>', 
        esc_url( $link ),
        esc_html__( $title, 'mydomain' )
    );

    return $output;

}, 10, 2 );

To disable the content pagination when wpse_show_all_content is true, we can use (PHP 5.4+):

add_filter( 'content_pagination', function( $pages )
{
    // Validate user input
    $show_all_content = (bool) filter_input( 
        INPUT_GET, 
        'wpse_show_all_content', 
        FILTER_VALIDATE_BOOLEAN 
    ); 

    // Disable content pagination
    if( $show_all_content )
        $pages = [ join( '', $pages ) ];

    return $pages;

} );

Hope you can adjust this further to your needs!

Update

As per discussion, here’s a way to target only a specific wp_link_pages() instance:

add_filter( 'wp_link_pages', function( $output, $args )
{
    // Nothing to do if 'wpse_show_all_link' is missing or not a true boolean string
    if( 
            ! isset( $args['wpse_show_all_link'] ) 
        ||  ! wp_validate_boolean( $args['wpse_show_all_link'] )
    )
        return output;

    // Validate user input
    $show_all_content = (bool) filter_input( 
        INPUT_GET, 
        'wpse_show_all_content', 
        FILTER_VALIDATE_BOOLEAN 
    ); 

    // Generate link
    $link = add_query_arg( 
        'wpse_show_all_content', 
        ! $show_all_content, 
        get_permalink() 
    );

    // Generate link title
    $title = $show_all_content 
        ? 'Show Content Pagination' 
        : 'Show All Content';

    // Append    
    $output .= sprintf( 
        '<a href="https://wordpress.stackexchange.com/questions/239209/%s">%s</a>', 
        esc_url( $link ),
        esc_html__( $title, 'mydomain' )
    );

    return $output;
}, 10, 2 );

where we introduce a custom wpse_show_all_link attribute that can take values like

1, 0, '1', '0', 'yes', 'no', true, false, 'true', 'false'.

Usage:

wp_link_pages( ['wpse_show_all_link' => true, ...] );