When adding a 'All Products' (which is a default WooCommerce feature) block to any page, the 'sold out' badge does not appear. It does appear in the default shop page.
Here's a screenshot of the default shop page where the products do have such a badge:
WooCommerce status report
### WordPress Environment ###
WordPress address (URL): https://rookieboards.store
Site address (URL): https://rookieboards.store
WC Version: 5.2.2
REST API Version: ✔ 5.2.2
WC Blocks Version: ✔ 4.7.2
Action Scheduler Version: ✔ 3.1.6
WC Admin Version: ✔ 2.1.5
Log Directory Writable: ✔
WP Version: 5.7.1
WP Multisite: –
WP Memory Limit: 256 MB
WP Debug Mode: –
WP Cron: ✔
Language: en_US
External object cache: ✔
Server Environment
Server Info: nginx
PHP Version: 7.4.18
PHP Post Max Size: 2 GB
PHP Time Limit: 1200
PHP Max Input Vars: 6144
cURL Version: 7.76.0
OpenSSL/1.1.1d
SUHOSIN Installed: –
MySQL Version: 5.5.5-10.3.25-MariaDB-log
Max Upload Size: 2 GB
Default Timezone is UTC: ✔
fsockopen/cURL: ✔
SoapClient: ✔
DOMDocument: ✔
GZip: ✔
Multibyte String: ✔
Remote Post: ✔
Remote Get: ✔
Database
WC Database Version: 5.2.2
WC Database Prefix: wp_
Total Database Size: 13.28MB
Database Data Size: 9.70MB
Database Index Size: 3.58MB
wp_woocommerce_sessions: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_woocommerce_api_keys: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_woocommerce_attribute_taxonomies: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_woocommerce_downloadable_product_permissions: Data: 0.02MB + Index: 0.06MB + Engine InnoDB
wp_woocommerce_order_items: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_woocommerce_order_itemmeta: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_woocommerce_tax_rates: Data: 0.02MB + Index: 0.06MB + Engine InnoDB
wp_woocommerce_tax_rate_locations: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_woocommerce_shipping_zones: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_woocommerce_shipping_zone_locations: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_woocommerce_shipping_zone_methods: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_woocommerce_payment_tokens: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_woocommerce_payment_tokenmeta: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_woocommerce_log: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_actionscheduler_actions: Data: 2.02MB + Index: 0.66MB + Engine InnoDB
wp_actionscheduler_claims: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_actionscheduler_groups: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_actionscheduler_logs: Data: 1.02MB + Index: 0.39MB + Engine InnoDB
wp_automatewoo_abandoned_carts: Data: 0.02MB + Index: 0.08MB + Engine InnoDB
wp_automatewoo_customers: Data: 0.02MB + Index: 0.09MB + Engine InnoDB
wp_automatewoo_customer_meta: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_automatewoo_events: Data: 0.02MB + Index: 0.05MB + Engine InnoDB
wp_automatewoo_guests: Data: 0.02MB + Index: 0.06MB + Engine InnoDB
wp_automatewoo_guest_meta: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_automatewoo_logs: Data: 0.02MB + Index: 0.05MB + Engine InnoDB
wp_automatewoo_log_meta: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_automatewoo_queue: Data: 0.02MB + Index: 0.05MB + Engine InnoDB
wp_automatewoo_queue_meta: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_commentmeta: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_comments: Data: 0.02MB + Index: 0.09MB + Engine InnoDB
wp_links: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_ms_snippets: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_options: Data: 2.09MB + Index: 0.08MB + Engine InnoDB
wp_postmeta: Data: 1.52MB + Index: 0.23MB + Engine InnoDB
wp_posts: Data: 1.52MB + Index: 0.06MB + Engine InnoDB
wp_sbi_instagram_feeds_posts: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_sbi_instagram_feed_locator: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_sbi_instagram_posts: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_snippets: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_termmeta: Data: 0.14MB + Index: 0.17MB + Engine InnoDB
wp_terms: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_term_relationships: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_term_taxonomy: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_usermeta: Data: 0.05MB + Index: 0.03MB + Engine InnoDB
wp_users: Data: 0.02MB + Index: 0.05MB + Engine InnoDB
wp_wc_admin_notes: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_wc_admin_note_actions: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_wc_category_lookup: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_wc_customer_lookup: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_wc_download_log: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_wc_order_coupon_lookup: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_wc_order_product_lookup: Data: 0.02MB + Index: 0.06MB + Engine InnoDB
wp_wc_order_stats: Data: 0.02MB + Index: 0.05MB + Engine InnoDB
wp_wc_order_tax_lookup: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_wc_product_meta_lookup: Data: 0.02MB + Index: 0.09MB + Engine InnoDB
wp_wc_reserved_stock: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_wc_tax_rate_classes: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_wc_webhooks: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_wc_zapier_history: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_woocommerce_shipping_table_rates: Data: 0.02MB + Index: 0.00MB + Engine InnoDB
wp_wpc_comments_subscription: Data: 0.02MB + Index: 0.05MB + Engine InnoDB
wp_wpc_phrases: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_wpc_users_voted: Data: 0.02MB + Index: 0.06MB + Engine InnoDB
wp_yoast_indexable: Data: 0.14MB + Index: 0.08MB + Engine InnoDB
wp_yoast_indexable_hierarchy: Data: 0.02MB + Index: 0.05MB + Engine InnoDB
wp_yoast_migrations: Data: 0.02MB + Index: 0.02MB + Engine InnoDB
wp_yoast_primary_term: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
wp_yoast_seo_links: Data: 0.02MB + Index: 0.03MB + Engine InnoDB
Post Type Counts
attachment: 157
customize_changeset: 2
custom_css: 1
global_product_addon: 1
jetpack_migration: 2
nav_menu_item: 18
oembed_cache: 10
page: 16
product: 24
revision: 90
shop_order: 2
wpautoterms_page: 1
wpcf7_contact_form: 2
Security
Secure connection (HTTPS): ✔
Hide errors from visitors: ✔
Active Plugins (21)
Akismet Anti-Spam: by Automattic – 4.1.9
WP AutoTerms: by WP AutoTerms – 2.4.5
AutomateWoo: by WooCommerce – 5.1.3
CoBlocks: by GoDaddy – 2.11.0
Code Snippets: by Code Snippets Pro – 2.14.1
Contact Form 7: by Takayuki Miyoshi – 5.4.1
Facebook for WooCommerce: by Facebook – 2.4.1
Smash Balloon Instagram Feed: by Smash Balloon – 2.9.1
Jetpack by WordPress.com: by Automattic – 9.7
Sold Out Badge for WooCommerce: by Charlie Etienne – 2.0.10
Woocommerce Product Page Social Share: by Sajjad Hossain Sagor – 2.0.2
WooCommerce Stripe Gateway: by WooCommerce – 5.1.0
WooCommerce Print Invoices/Packing Lists: by SkyVerge – 3.10.0
WooCommerce Product Add-ons: by WooCommerce – 3.3.0
WooCommerce Sequential Order Numbers Pro: by SkyVerge – 1.17.0
WooCommerce Shipment Tracking: by WooCommerce – 1.6.26
WooCommerce Table Rate Shipping: by WooCommerce – 3.0.30
WooCommerce Zapier: by OM4 – 2.0.7
WooCommerce: by Automattic – 5.2.2
WooDiscuz - WooCommerce Comments: by gVectors Team – 2.2.9
Yoast SEO: by Team Yoast – 16.2
Inactive Plugins (0)
Dropin Plugins (2)
advanced-cache.php: advanced-cache.php
object-cache.php: Memcached
Settings
API Enabled: –
Force SSL: –
Currency: EUR (€)
Currency Position: left
Thousand Separator: .
Decimal Separator: ,
Number of Decimals: 2
Taxonomies: Product Types: external (external)
grouped (grouped)
simple (simple)
variable (variable)
Taxonomies: Product Visibility: exclude-from-catalog (exclude-from-catalog)
exclude-from-search (exclude-from-search)
featured (featured)
outofstock (outofstock)
rated-1 (rated-1)
rated-2 (rated-2)
rated-3 (rated-3)
rated-4 (rated-4)
rated-5 (rated-5)
Connected to WooCommerce.com: –
WC Pages
Shop base: woocommerce/woocommerce-blocks#206 - /shop/
Cart: woocommerce/woocommerce-blocks#6 - /cart/
Checkout: woocommerce/woocommerce-blocks#7 - /checkout/
My account: woocommerce/woocommerce-blocks#8 - /my-account/
Terms and conditions: woocommerce/woocommerce-blocks#256 - /wpautoterms/terms-and-conditions/
Theme
Name: Storefront
Version: 3.5.1
Author URL: https://woocommerce.com/
Child Theme: ❌ – If you are modifying WooCommerce on a parent theme that you did not build personally we recommend using a child theme. See: How to create a child theme
WooCommerce Support: ✔
Templates
Overrides: –
Action Scheduler
Complete: 2,113
Oldest: 2021-04-09 10:41:14 +0000
Newest: 2021-05-10 10:31:40 +0000
Pending: 5
Oldest: 2021-05-10 10:32:40 +0000
Newest: 2021-05-10 19:50:11 +0000
WooCommerce Zapier
Database Version: ✔ 12
Detailed Logging Enabled: No
Legacy Mode Disabled: ✔ Yes
SSL: ✔ Yes
REST API Authentication Key(s): ❌ 0
Task History Record Count: ❌ 0
Status report information
Generated at: 2021-05-10 10:31:44 +00:00