Why not just drastically reduce the price of the 360 to be an entertainment device. It has all the same apps and services afterall.
Cost to maintain the older parts? It's much cheaper to just make a stripped down XB1 that uses, say, 80% of the same parts as the main XB1 than to keep the different production line for the 360 going.
Anyways, with the ability to go digital for even AAA games, MS selling a XB1 "lite" as a pure media box but with the ability to allow people to download a AAA game if they wanted to sounds good to me. It's working the market from the other way, actually from both directions at once. You have the main XB1 that works from core gamers towards casual+media and you have a XB1 lite that works from media+casual towards core gaming for those who choose to give it a try.
Sounds like a win win to me, specially if these are at the price point where they can be a set-top box option through a cable service provider.