Naming
"It seems to me that the 'Azure' brand is being attached to far too many things of dubious benefit."
(Not getting into the "dubious benefit".)
This is typical of Microsoft though. When .NET (the common runtime environment) was first released, they had planned to call their single sign-on service .NET, and had either released or planned to release Office .NET, Visual Studio .NET, Windows Server .NET, I think Exchange .NET... just off the top of my head. I think they were even planning to stick the ".NET" name onto DirectX somehow. Some marketing type at the company will decide to stick "name of the day" on all their products, later on they'll realize it's way the hell to confusing and stop. Apparently today the "name of the day" is Azure.
I have to admit, if hosted Office's performance characteristics make it tempting to get a dedicated connection to get decent performance, I'd probably instead conclude the performance isn't good enough and not use it rather than spend the big bucks on some direct line.