We've had good results with the forwarding that Emily posted. It's totally optional if you want to keep a copy of the forwarded messages. It can be a helpful auditing tool if some users are worried not all messages are making it into Genesys Cloud.
I don't think you would want to have it done client side. You would only want to enable it on accounts like
sales@acme.com or
support@acme.com that you want handled by agents.
Also the "registering of the domain" can be confusing. When you add a domain name Genesys Cloud will ask you to add a CNAME record with some random text to your domain. This will prove that you own the domain and Genesys will be comfortable sending outbound emails using your domain.
It is also important to enable DKIM Authentication, without this some customers may receive a warning that "the actual sender of this message is different than the normal sender"
Original Message:
Sent: 05-07-2021 06:58
From: Emily Kammerer
Subject: Email routing (keeping your mailboxes and redirecting inbound messages) - how are you do it?
Hi Jeff
We've got this set up as a Forward:

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Emily Kammerer
Ascendium Education Solutions, Inc.
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