LaurentF1000 regarding the first one
Why can’t we uninstall our last Exchange server?
Uninstalling your last Exchange Server will result in various configuration changes including the Exchange Organization container getting removed from Active Directory, which will make it impossible for our management tools package to manage Exchange attributes.
That means in exact that state (removed Exchange configuration from Active Directory) it's impossible for the management tools package to manage Exchange attributes. Because the attributes don't exist then.
But that lower part in the text is not contradicting:
Our organization never installed any Exchange Server 2019 servers. Can we use this solution to manage our Exchange recipients? Do we need to install Exchange 2019 Server first to do it?
Exchange 2019 Management tools can be installed in any organization that currently has Exchange Server 2013 or newer version. You do not need to have a running Exchange Server 2019 before you can use the management tools. Simply Prepare Active Directory for Exchange Server 2019 (using the 2022 H1 Exchange Server 2019 CU or newer) and then install the Management tools on a domain joined machine.
The important part is the last sentence of that (underlined). There is an own question that should clarify this further:
We removed (uninstalled) the last Exchange server long ago. Can we now install the updated Exchange Server 2019 Management tools to help us manage recipients if we use Azure AD Connect?
Yes. Simply Prepare Active Directory and install the Exchange 2019 Server Management tools on a domain-joined machine.
So if you don't have the Exchange Schema extensions in your Active Directory - doesn't matter if you ever had them before - just Prepare the AD Schema and you're good to go.
//CC fx1333X I think this should also clear up your concerns.