Since Publisher 2021 was originally installed on this computer, I don't think you need to keep searching the backup drive for PUB.EXE or copying the old Office files. Those files by themselves won't recreate a working Publisher installation.
Before buying another computer or another copy of Publisher, I suggest checking whether your original Publisher 2021 license is still associated with your Microsoft account.
Sign in to your Microsoft account > Services & subscriptions using the Microsoft account/email address you were using when Publisher 2021 was originally installed. Look for Publisher 2021, Office 2021, or another Office 2021 product in the list.
Once you redeem an Office 2021 product key, the license is associated with your Microsoft account, and you normally reinstall it from that account rather than finding and re-entering the original product key.
If you find Publisher/Office 2021 there, don't install anything yet. Let us know the exact product name displayed (for example, Publisher 2021, Office Professional 2021, etc.). That will tell us what license you actually have and the safest way to reinstall it.
If nothing related to Office/Publisher 2021 appears, try any other Microsoft account/email address you may have used around 2021. The full Office product key generally isn't displayed inside Office, so not finding the key written down doesn't necessarily mean you've lost the license.
As for buying another Windows laptop specifically for Publisher 2021: don't spend the money yet. Let's first establish whether you already own a reinstallable Publisher 2021 license.