Unfortunately, if the owner of the content is not interested in keeping this site up, the content will be lost sooner or later anyways. He probably also does not bother to install security updates and he will most likely stop paying the bills at some point (domain name, hosting server, etc).

Installing LetsEncrypt is not much work and he might be motivated if a lot of people ask him. If he is really not interested, it is probably best to archive the website to a real archive and hope they make sure they content remains available. Unfortunately this also means that the archive will no longer be found on google or most other search engines. It is really a shame that there is no work on google's side to make sure archived content can be found among other search results.