Pages does not automatically save documents to computer or iCloud

If you want Pages autosave to work, then System Settings > Desktop & Dock > Ask to keep changes when closing documents must not be enabled, and you should initially manually save your blank document with a valid name to enable automatic autosave for additional document content. If you do this, the odds of your losing document content owing to Pages freezes or power outages is minimal to completely avoidable.

You will know that autosave is disabled because a black dot will appear in the red traffic light of your Pages document titlebar until you manually save the document, and then reappears with new content, before the next manual save. You will never see this with autosave enabled.

Don't say the "latest software on everything" as many people over the years have said this to save time and turned out to be wrong. We need explicit versions mentioned here to help us help you. In this case, it wasn't necessary.

I can assure you that autosave works just fine here with Pages v14.4 and Tahoe 26.1 where that Desktop & Dock setting is disabled. No change for me from Sequoia, Sonoma, Ventura, etc.

There is no button on Microsoft Word for autosave (only to OneDrive) either. It is user enabled in its Preferences > Save panel. Same silent background behavior as one experiences in Pages.

Yes, Apple could simplify this by placing a Pages-specific "Enable Autosave" checkbox in Pages Settings, but with the Desktop & Dock placement, other Apple applications are able to detect whether to use or not use autosave (e.g. Preview).

You can always use the Provide Pages Feedback menu item from its application menu.

The shift-key procedure is only needed after Pages hangs, or a power outage.

I suggest that you shutdown your Mac, not just reboot. When you have powered it back on, press and hold the shift key while launching Pages. This will prevent it from loading its last saved execution state (hang), and it may resolve the issues you are experiencing.

Otherwise, it may be something that you have installed that is conflicting with the operating system and/or Pages itself. Remove any and all anti-virus and so-called "cleaning applications" as they have caused users issues for years.