How do I programatically insert a new menu item?

Before being printed, all the menu items get run through a filter. You can target the wp_nav_menu_items filter to tack things on to the menu:

// Filter wp_nav_menu() to add additional links and other output
function new_nav_menu_items($items) {
    $homelink = '<li class="home"><a href="' . home_url( "https://wordpress.stackexchange.com/" ) . '">' . __('Home') . '</a></li>';
    // add the home link to the end of the menu
    $items = $items . $homelink;
    return $items;
}
add_filter( 'wp_nav_menu_items', 'new_nav_menu_items' );

Or, to be more specific, you can target only the desired menu by replacing the add_filter line from above with the following, and replacing $menu->slug with your menu’s actual slug name:

add_filter( 'wp_nav_menu_{$menu->slug}_items', 'new_nav_menu_items' );

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