There is a bug in FFXIV in their device enumeration. Everytime a change is happening in your PC your controller will get disconected. You can prove that by plugging a USB Drive while playing FFXIV and you will get your controller disconnected.

This is not an issue with your PC, it really a bug in their implementation. Luckily there is a work around.

The work around is not a fix, it may make some of your device failed to work.

You can get more info here : https://www.youtube.com/watch?v=nXSE9Rd8QxU
Here is my video of the issue : https://www.youtube.com/watch?v=VsOdhha6efA

Honestly I find it sad that SE is not agreeing that this is a bug. I spent hour trying to figure out what was wrong.

One last thing, if the above solution does not work, You can let the device manager open while playing and you will see that it will refresh when your controller is getting disconnected, this mean a device was changed. It can help you to find which device it is...