Hi Nicolas,
I understand that this field is related to old JMS Multisites logic. We don't mind the legacy code, but there is a clear inconsistency in how it behaves now.
The issue is that the category listing and the product page use different logic for the same data. The listing shows the lower price (ignoring the Site ID), while the product page filters it out. Even with 'Show several prices' enabled, the array is truncated before reaching the template.
If the core logic treats a non-empty price_site_id as a strict priority filter even when no multisite environment is present, it effectively breaks the price display settings for many users. We'll find a workaround on our end, but you might want to check why the fetching logic differs so much between the listing and the product view.