Does a web service exist for performing management tasks against Exchange 2013 - equivalent to powershell commands such as:
Enable-Mailbox
Add-MailboxPermission
Set-Mailbox
...
I am aware of options like wrapping remote powershell code etc, but from a development standpoint that is a terrible way to integrate. I am also aware of the Exchange Web Service for creating clients (like OWA), but that does not cover administrative tasks.
I have googled and found nothing, so I assume there is nothing better for Exchange 2013 - but I would like it confirmed. :)
No, that does not exist. PowerShell is the Primary way of interacting with Exchange Management. It is however fairly easy to use PowerShell in a .NET application. After all, it's "just" an API then...
As you already noted, EWS does not contain Management Tasks, neither does the new REST API, or any other API that I know of.