I did manage to control my Apple TV (currently running tvOS 9.2) from a python script. It turns out that you don't need to use Home Sharing to have a remote app control the Apple TV. I don't know if the following method will work if Home Sharing is enabled, but with it disabled on the Apple TV, the iOS Remote app has the option to manually add a device. (This may require removing all of the devices it is already paired with, since that was unfortunately necessary for me to get it to display the 'Add a device' button.) Once I had paired my iPhone to the Apple TV, I recorded some of its requests, copied the pairing GUID, and then constructed some of my own requests.
The only three requests necessary to make are:
/login?pairing-guid=< your pairing guid here >&hasFP=1
Logs into the Apple TV. The last four bytes of the response's is a session id, encoded as a big-endian four byte integer.
/logout?session-id=< your session id here >
Logs out. Not strictly necessary, as I found that logging in simply gets you a new session id, but probably not a bad idea to do things the way it expects.
/ctrl-int/1/controlpromptentry?prompt-id=114&session-id=< your session id here >
Send user input to the Apple TV. The data is one of several buffers that input a command, or possible a moving touch. For movement in the cardinal directions, sending several of these requests to simulate a moving touch is necessary.
I have a python script demonstrating how to do this here: http://pastebin.com/mDHc353A
Utilizes the requests library: http://docs.python-requests.org/en/master/
Also special thanks to Adam Miskiewicz / github user skevy, since I made use of this file in his atlas-backend repo that conveniently had the right buffers to send for movement: https://github.com/skevy/atlas-backend/blob/master/atlas/services/appletv.coffee