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Thanks for the playing! There are 2 potential culprits on the xbox360 controller issue but since it was working and then suddenly started quitting to menu my guess is you might have Steam or Steam Big Picture open/running  (which for some reason likes to obliterate/override  control schemes in certain games, ours currently included). Turning those off fixes the problem. If it isn't related to Steam I would suggest downloading the most recent 360 drivers, modern windows doesn't bundle 360 controller drivers in regular updates as they want everybody using xbone controllers so at times it is necessary to go and manually download them. 
If neither of these things fixes the problem, please do let me know so I can note the problem on the page until some kind of fix is found.
The boss checkpoint placement is indeed meant to give a little breather but in some cases that time can also be used to mount a preemptive attack on the boss before the fight begins. 

Thanks for the reply! I think you're right about the issue being drivers or Big Picture mode, I'm a bit scared to change anything in Steam since I finally got the DS4 working for Final Fantasy 15 with the correct button prompts. I remembered I had a wired Xbox One controller for the Steam Link in another room so plugged that in and after configuring the controls it worked fine. So much better playing with a controller, maybe I'll even beat the second boss! I love the feeling of having to move into danger to restore your health, it gives it such a unique feel compared to other shooters. I find myself trying to barrel roll back in the opposite direction too soon which is currently the main thing getting me killed.