In theory, not a problem - this is was a Call Routing Announcement is designed to do.
However, the greeting length is going to be an issue. Whenever a greeting is playing, a voicemail port is tied up. When all ports are in use, you have no voicemail functionality: auto attendants, mailboxes, announcements, etc. will not work.
As a general rule, greetings are kept as short as possible to prevent busy ports.