Can I download Pages on Mac Mini without it pre-installed?

If you've downloaded Pages in the past, you'll be able to download a compatible version from the App Store by going to the Purchases section.

Otherwise, if you have access to another Apple device that is compatible with the latest version of Pages, use this device to download Pages. You may then be able (but I'm not 100% sure) to download Pages on your Mac mini by going to the Purchases section in the App Store.

Your profile shows a Mac mini with macOS v13.1 (Ventura) and you mention the Mac mini in question only supports Catalina (10.15.7). Can you clear up that contradiction?

A Mac mini running macOS Ventura can directly download Pages from the Mac App Store on that Mac since Pages v14.1 requires Ventura or later for installation. It would be wise if Ventura is involved in using Software Update to install the last update to 13.6.7.

However, if Catalina is involved, and as has been stated by 6x6 about the condition of installing it previously from the Mac App Store on any other Mac (by you) with the same Apple ID, then you can still get Pages v11.1 for Catalina.

If the last paragraph is true, then launch the Mac App Store, sign in with the same Apple ID, and then press cmd+0 (zero) to open the Purchased panel. If Pages appears there, you can click the appropriate indicator to install it. There may be one or two dialogs, neither of which do you click Cancel. The final dialog will offer to install the last compatible version (11.1) of Pages for Catalina.

Apple provides no other means to obtain past full versions of Pages, Numbers, or Keynote than the last paragraph.