This is Elias from Shopify Support. Thank you for reaching out.
I understand you've adjusted your inventory as 0, but the sold out message is not being shown on your homepage. It's possible that you may have "Continue selling when out of stock" enabled for your products, which allows the products to be sold regardless of your inventory levels. If this feature is turned on, then the sold out message may not appear for some themes. In order to check this setting, you can edit the products or use the bulk editor.
Here's how the setting looks when editing products or variants individually:
Here's how you can use the bulk editor to check this feature on multiple products:
Could you check if this might be what's causing the issue for you?