No rumble feature

Yes. No rumble feature at all

I remember gaming before rumble features, the n64 was the first console to use it with add on rumble packs!! And I loved it ever since the first time I expierienced it. Even back then you had the option of rumble or not. I agree it should be a feature of
the controller now days. I would hardly ever turn off though.

Controllers disconnecting this way can sometimes be cause by the rumble feature in the pad. If it is too severe and the battery is a little loose it can break the contact and turn off the pad.

Try turning off the rumble feature and see if it happens again.

When battery is low, the rumble feature is disabled and the icon light blinks.

You can on the 360, just press the guide button and go over to control settings in the guide menu. You should have the option to disable the rumble features. From what I have heard you currently can not turn the rumble feature off on the Xbox one.