No, you don’t need to keep your on-premises Exchange Server unless you need it for mail relaying, managing public folders, or other legacy stuff.
Your Office 365 environment can be managed using Easy365Manager without Exchange on-premises.
Besides on-premises Exchange, Easy365Manager also replaces the need to continuously log in to the various web consoles and use PowerShell for various tasks like calendar delegation, Outlook auto-mapping, and Azure AD Connect synchronization.
Read our recommendations on safely shutting down your last on-premises Exchange Server.
You can also keep your on-premises Exchange Server and run it side-by-side with Easy365Manager.