I don't really see what has to be the big issue with adding the ability to alt-tab.
I know it is a bit of extra coding, and maybe a (small) redesign of their graphics engine (depending on how it works atm). But it really wouldn't hurt for them to add it.
Hell even my home-made puzzle game from 10 years ago doesn't have any problem with it. Well it did at first, but took me like a day or so to fix that after it started to annoy me. (didn't have decent tutorials back then ).
That same could will also fix crashes based on UAC requests made by the system because of some software wanting to update on your system.
So unless they have cross-platform egine code that doesn't allow for platform specific features, I don't see what the big problem for it would be.

In meantime, I just run it in Windowed Mode (1920x1080) on a resolution of the same size, and use a little program to reposition the window so it fills the screen (as windows won't allow you to do that by hand)
Works good enough, with only a little performance loss.

Another thing you can do is, launch FFXIV from within Steam, and use the Steam overlay and it's browser to get on the internet. (if you prefer to use fullscreen)