We are looking at using Office 365 groups instead of a shared mailbox. Here are the issues we can't seem to overcome.
1. You can't import historical email into a group so everthing in the past has to stay in a shared mailbox.
2. You can't create subfolders everything stays in the inbox
3. when a user has send as or send on behalf of persmissions for the group and they reply to the message by default its coming from the user not the group. We need it to default to the group so we can avoid the extra clicks and erros that will come from careless group members. :)
Groups are not a valid replacement option for a Shared mailbox, especially one that is used in workflows. There are other limitations that the ones you mentioned which make them unsuitable choice, for example Rules, or controlling sent/deleted items, etc.
But you can actually migrate content to Groups. There are 3rd party tools that do that, or you can use OWA as a workaround (it allows you to add the Group as additional mailbox and copy/move content to it).
You can move email from your shared mailbox to an office 365 group but it's painful. First move, the email to a folder in your office 365 account. Then move the email to the office 365 group. This can only be done in Outlook OWA and the limitation is you have to move one message at a time. Hopefully that will be fixed soon.