If you are using Time Machine backing up to an external hard drive on the 2013 Mac, once you have the new Mac, you can go through the initial setup, creating a user account (same username and password as on your old Mac). Then visit System Settings > General > Software Update and apply any macOS Tahoe 26.* updates that are available.
Then, you turn off Time Machine on the new Mac via Settings > General : Time Machine : Options : Backup Frequency : Manual. Now you mount the Time Machine drive from your old Mac. You have prevented the new Mac from backing up to your old Big Sur TM drive which you mount next.
Now, you can run Migration Assistant (on the Desktop, press shift+cmd+U and double-click Migration Assistant) which will detect your mounted Big Sur Time Machine drive that you select as the source for the migration. Do not select the Applications category when presented by Migration Assistant. When it is done counting up each of the other categories, you turn it loose and it will migrate documents and settings from Time Machine to the new Mac. Your Pages and PDF documents will be where they were on the 2013 Mac. See Transfer to a new Mac with Migration Assistant - Apple Support
When the migration is done, unmount the Big Sur Time Machine drive and set it aside. I would not use it for Time Machine backups on a new Mac.
At this point, you can visit the Mac App Store, sign-in with your Apple ID, and search for Pages, which will reveal the current Pages, Numbers, and Keynote for download. These should open your migrated documents without any issue. You can continue to view your PDFs in Apple's Preview.
I strongly recommend that you acquire a new Time Machine drive to back up the new Mac to Time Machine. I happen to use 2 TB Crucial X9 SSDs that I format with Disk Utility as APFS. Otherwise, anything from OWC would be OK too. My naming convention is TM9_Tahoe for the drive name. A rule of thumb for TM drives is that they should be at minimum, twice the capacity or more of the internal Mac drive. Once you have mounted the new TM drive, revisit System Settings > General > Time Machine and Click the + symbol to add your TM drive. Then under Options, change the backup schedule to Automatically Every Hour.
