For the secondary server, a mail server is necessary for cronjobs or form mails, but it doesn't have to communicate with the internet.
In this case, Postfix as a satellite system is suitable.
# apt install postfix
The firewall doesn't have to be opened, since this satellite system is for the local system only.
This is all for the satellite system. Postfix will receive the mails from local systems, and forward all of them to the relay host.
The relay host (primary server) has to accept the connection from the satellite system. Add the IP address to the mynetworks in /etc/postfix/main.cf to handle the satellite server same as the localhost.
mynetworks = 127.0.0.0/8 [::ffff:127.0.0.0]/104 [::1]/128 IPv4 [IPv6]
How to receive the mails for the satellite server (i.e. cron results) depends on your policy. In this case, the mails sent to the satellite server accounts will be delivered to the main server mailbox.
To receive the mails to the satellite server, add the domain to virtual_mailbox_domains in /etc/postfix/main.cf.
virtual_mailbox_domains = main.example.jp satellite.example.jp
Reload Postfix after you edit main.cf
# systemctl reload postfix
Add the mail addresses to /etc/postfix/virtual to consolidate system accounts to 'info'.
email@example.com firstname.lastname@example.org email@example.com firstname.lastname@example.org email@example.com firstname.lastname@example.org email@example.com firstname.lastname@example.org
Use postmap to renew the database.
# postmap /etc/postfix/virtual