We don't know the laws from all countries. Most of our customers use our software to recover a site that they created themselves, which is probably legal everywhere.
If you are not the original creator of that site, then it's probably illegal in some countries to recover the content of a website.
However, for most websites the copyright is credited in the footer to the domain, such as "Copyright  example.com" . So if you own the domain, you could make a case for owning the copyrights.
It's a grey area for sure, and there is probably nobody who can tell you with certainty, since there is not much legal precedence.
For the expired content, it's unlikely anybody will every find out or care. We make sure the content is not published elsewhere on the internet, so it will be difficult for the other party to provide evidence of financial damages.
In reality: most content on the Internet is duplicated elsewhere, so it's unlikely that these type of borderline cases will cause you any problems. The worst case scenario is receiving an official DMCA takedown request. You then have to remove the content before a certain date to avoid legal action. If you use our software to create a Private Blog Network (PBN), and you are afraid of legal problems, then we recommend removing or replacing the original contact details (phone number, street address).
We never heard from a customer who actually got into legal troubles because of using our services.