According to this 2016 gameranx video:
The US military actually uses an Xbox 360 controller for EOD robots - which are essentially robots that pick up bombs, or deliver bombs.
(EOD stands for Explosive Ordinance Disposal.)
Is that true?
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The controller can be seen around the 2:42 minute mark of this B-roll footage from iRobot's media kit.
Furthermore, the following picture (taken from the PackBot Wikipedia page, and provided by the U.S. Navy before that) clearly shows a 3rd-party console controller in the bottom-right corner:
In short, they may not be able to "plug in any Xbox 360 controller", but they certainly use similar-looking controllers and, with the proper ruggedization, may even be able to use off-market options.