The Exchange hybrid deployment feature allows for the coexistence of Exchange mailboxes both on-premises and in Microsoft 365. Azure AD Connect synchronizes a specific set of attributes from Azure AD back into your on-premises directory. You can turn this on at any time.
Here are the attributes synced once enabled:
and written back to on-prem:
Note - its not mandatory to have this enabled. Exchange Hybrid configuration isn't really required unless you are going to host mailboxes in both locations at the same time, and thats normally during the initial setup where you are migrating users.
There is also the caveat that you are introducing an Exchange Server into your on-premises environment. While its purely going to be used for management as per your scenario, it still introduces the same admin over head of maintaining, patching, setting up connectors, security concerns (and there have been a lot lately....) around Exchange Servers.
When you say you want to get this to a "supported state" - to me, this looks like the state you should want to be in as you have no requirements or dependencies on any on-premises infrastructure apart from Azure AD Connect. What I would be asking is: could you perform the management tasks you are looking to do from Exchange Online?
Hope this helps - note that this is my opinion only!
- If the reply was helpful please upvote and/or accept as answer as this helps others in the community with similar questions. Thanks!