If you were on Microsoft Windows, you would expect the file extensions to be hidden by the operating system. However on macOS, file extensions help the operating system determine the default opening application, and others that are compatible in opening the document.
When I deliberately hide the Pages document extension in the Finder Get Info panel, and then open that document with Pages v14.4, the file extension remains hidden during changes to the document, saving, and closing the document. This is the tested case on macOS Sequoia, and again on macOS Tahoe 26.0. Pages simply honors the existing extension visibility status as it is an extended attribute.
Many versions of Pages ago, one had an option on the Save/Save As panel in Pages to override the extension visibility status during a document save. That has long since been removed. There are no adjustments in the Finder Settings to force a default hidden document extension, though you have found the sole capability in the Get Info panel that operates on a per-file basis.