Ah sorry I forgot they use xbox wireless now instead.

It's a higher bandwidth but from what little information Microsoft has released about it, it doesn't appear to be lossless and does use compression for the audio transfer which might negatively impact sound.

So, you should see quite an improvement to the gx with an amp. I can't find the specifications for the controllers 3.5mm output but being so cheap and small I'd be surprised if it output anything over 1mw.

Having to use them at max volume greatly increases the chance for distortion. So an amplifier will deliver the necessary power to make the gx sing. I'd still personally recommend a dac as well (or a combo unit) just so that you have as clean a chain as you can, after all those headphones can be quite revealing and adding an amp to the controller output might exacerbate discrepancies in sound.

Depending how much you want to spend, a fiio k5 pro would be a pretty affordable little unit.

Also, if your TV doesn't have optical out, you can use a hdmi splitter to create your own. No usb or wireless dacs work with the xbox as far as I can't tell.