Create new email alerts for board members

As I understand the issue, the ask is to notify the board members that they have email to read instead of forwarding a copy of the messages from their hosted mailbox to their personal mailbox. Forwarding email outside Office 365 to personal mailboxes is always a bad idea when it comes to company officers and executives because of the danger of compliance violations. I would therefore not look at any forwarding solution.

It's reasonable and in line with the obligations that board members take on when they join a board to ask them to sign in to read confidential information and to only store that information within a repository managed by the company. It would be possible to write code with EWS or the Graph APIs to monitor mailboxes and send board members a summary of messages waiting their attention (PowerShell is out of the question because PowerShell can't access user content, which is why EWS and the Graph exist). 

Alternatively, if you want an OOTB solution, set up Teams and and create a team for the board members. They'll receive email notifications when messages are posted that need their attention. I use Teams for board-level communications in companies I work with and it's a nice solution because boards are small groups that need to share confidential messages and documents. Teams does this OOTB.