Hi Aj,
An ERMS E-mail is a workflow task item and therefore shown in the Home Screen when logging on in a SALESPRO role. That's the same for me as well.
To find out why the IC agent is not able to find it in the agent inbox, you might check KBA 1846041 How to analyze why an ERMS inbound E-Mail is not found in the agent inbox and note 1464849.
Not sure if you use BADI solution from note 1375170, if yes then the assignment of servicde UT_WORKITEMTEXT after service RE_RULE_EXEC in the service manager profile is very important.
Check if in transaction SWI1 if the agent is assigned under agents in the work item, if , check if the service manager profile has assigned
Best Regards,
Sigrid