[Q] Best 3rd party controller out-of-box topic

I have some games on my Fire TV I'd like to play, but need a game controller. I don't currently have any compatible controllers, so I thought I might as well buy the official one. However, it looks like both it and the Nyko pad are on back-order for around a month. Rather than wait, I'd like to get a decent third party controller, preferrably wireless. Can anyone say definitively which controller (other than the official one or Nyko) is the most functional out-of-box. FYI, my Fire TV is unrooted with current firmware, so anything requiring root is out.

Here are some of my current choices:

Microsoft Xbox 360 controller for Windows (included USB receiver)
Logitech F710, wireless USB receiver
Logitech F310, wired USB

The Xbox receiver supposedly supports up to 4 controllers. Would the Fire TV and any apps currently out support them? I'm partial to Logitech gear, but I read somewhere that on the Fire TV the F710 uses "xinput mode", emulating an Xbox controller. Does this mean it would work just as well as the Xbox or a close second? While I prefer wireless, I listed the F310 for the obvious benefit of not having to recharge or replace batteries. I don't play enough to where a cord would really bother me. If there are major caveats with the wireless ones, I might just get the F310.