How can I quickly get a system showing the “Uncaught Error: Class ‘WP_Site_Health’ not found in” up and running again?

Have a look at what point in your code you’re making calls to the WordPress REST API or any other code which makes use of WP_Site_Health.

If you’re making calls in your <theme>/functions.php file, for example, it won’t work because functions.php is included before class WP_Site_Health in wp-settings.php.

See wp-settings.php code (WordPress 5.7.2):

// Load the functions for the active theme, for both parent and child theme if applicable.
foreach ( wp_get_active_and_valid_themes() as $theme ) {
    if ( file_exists( $theme . '/functions.php' ) ) {
        include $theme . '/functions.php';
    }
}
unset( $theme );

/**
 * Fires after the theme is loaded.
 *
 * @since 3.0.0
 */
do_action( 'after_setup_theme' );

// Create an instance of WP_Site_Health so that Cron events may fire.
if ( ! class_exists( 'WP_Site_Health' ) ) {
    require_once ABSPATH . 'wp-admin/includes/class-wp-site-health.php';
}
WP_Site_Health::get_instance();

Despite the lack of elegance, one thing you can do about it is to ensure the class is loaded by copying part of the code from wp-settings.php and pasting it before your code. For example, in your <theme>/functions.php, you can do this:

if ( ! class_exists( 'WP_Site_Health' ) ) {
    require_once ABSPATH . 'wp-admin/includes/class-wp-site-health.php';
}
WP_Site_Health::get_instance();

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code depending on WP_Site_Health
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