Thanks for the information.
What is the DNS server setting of the NIC on your Exchange server?
The preferred DNS server should be domain controller and the alternative server should be left blank.
And since rebooting the server moves the messages to sent items, please also search the message tracking log and see if you can see the delivery process:
Get-MessageTrackingLog -sender <sender mailbox> -recipients <recipient mailbox> -messagesubject <subject>