Progmatically adding menu links to the default (Top) or Footer menu

You have to register the menu location first.

Also, I've added else statement to be sure $menu_id is defined when menu exists. Otherwise, you'll see the PHP Notice about undefined variable. WP_DEBUG must be turned on when you develop, that's why you don't see the notice.

// Initialize name and location
$menu_name     = 'Simple Inventory Menu';
$menu_location = 'simple-inventory-menu';

// Register the location:
register_nav_menus(
    array(
        $menu_location => $menu_name,
    )
);

$menu_exists = wp_get_nav_menu_object( $menu_name );

// If it doesn't exist, let's create it.
if ( ! $menu_exists ) {
    $menu_id = wp_create_nav_menu( $menu_name );

    // Set up default menu items
    wp_update_nav_menu_item( $menu_id, 0, array(
        'menu-item-title'   =>  __( 'Home', 'textdomain' ),
        'menu-item-classes' => 'home',
        'menu-item-url'     => home_url( '/' ),
        'menu-item-status'  => 'publish',
    ) );

    wp_update_nav_menu_item( $menu_id, 0, array(
        'menu-item-title'  =>  __( 'Admin', 'textdomain' ),
        'menu-item-url'    => home_url( '/index.php/admin/' ),
        'menu-item-status' => 'publish',
    ) );
} else {
    $nav_menu = get_term_by( 'slug', $menu_location, 'nav_menu' );
    $menu_id  = $nav_menu->term_id;
}

if ( ! has_nav_menu( $menu_location ) ) {
        $locations = get_theme_mod( 'nav_menu_locations' );
        $locations[ $menu_location ] = $menu_id; // no PHP notice here now
        set_theme_mod( 'nav_menu_locations', $locations );
}

You can expand the code to clean up modified 'nav_menu_locations' on the plugin deactivation. See register_deactivation_hook.

Complemented according the OP comment

If you already have registered menu locations (depends on the theme and navigation-menu-related plugins you use), and want your menu to appear there, just change the $menu_location value to the existing location (for example, 'footer-nav' or 'main-nav') and skip the register_nav_menus() step.

The downside is that this code will not work with any theme because of naming discrepancy. To make the code universal it have to be rewritten according to the mature requirements specification.

NO! Don't do the exposed above!

You'll have a lot of problems. Wait for me to get the right way. I also know where to utilize the menu you are talking about. That's why I'll develop it till them copmletely working solution.