Hi @CompactRaptor,
If you created it with Certbot, the items in /etc/letsencrypt/live are symbolic links (wihch other operating systems call “aliases” or “shortcuts”) to other files in /etc/letsencrypt/archive. Copying only items from /etc/letsencrypt/live to a new location without also copying /etc/letsencrypt/archive to the corresponding location won’t work for this reason.
We structured it this way because Certbot is meant to be used with automated renewals, so it keeps track of various things that are used for this purpose within /etc/letsencrypt. Then running certbot renew should automatically update all of your certificates by reissuing ones that are near expiry (although this feature isn’t available for wildcard certificates if you used --manual and created DNS records manually).
If you want to copy the individual certificates and associated files over manually (and you’re willing to repeat this process whenever you renew the certificates), you could find them in /etc/letsencrypt/archive. If you want to copy the whole structure, consider using rsync -a to copy all of /etc/letsencrypt onto the other machine (then the symbolic links will work).
Alternatively, maybe you could run Certbot on the AWS server itself rather than on a separate machine. In many configurations, this will produce a nicer experience because Certbot can create and update the certificate files on the same machine where they’re going to be used.