For starters, the Xbox One's backwards compatibility can make the 360 hardware obsolete, depending on what you intend to play. I sold my 360 a month or so after I got my One, as I knew I wouldn't take the time to play both. I can't say I miss the thing. The real reason I think it's time to let the 360 go, is if you're wanting the latest games to be their best. Shadow of Mordor had the best aspect of the game, it Nemsis A.I. system, gutted for the last-gen consoles, and the quality suffered (look at its PS4/X1 ratings vs PS3/360). Black Ops 3 is going to be $10 cheaper on the last-gen consoles than the current ones because they're not putting the campaign in AT ALL. Dying Light, the best game I've played on Xbox One, didn't even go to the last-gen consoles. Then, you've got the latest from 343 and Turn 10 (Halo and Forza), some other nice exclusives (Sunset Overdrive, Rare Replay), and the upcoming third-party stuff that's going to abandon the 360 (like Ubi's Division and Siege).
If you're more about playing older stuff because you don't like spending $60 to get a game at release, I'm sure there are a lot of hidden gems from the console you can still enjoy. However, 2015 is really the start of the end of the 360's major software support. Microsoft's IP is barely bothering to support it. Third-party stuff is on its way out. I think this could legitimately be the last CoD on the 360, unless they want to continue releasing gutted versions with online play only. For new games, the 360's dead in the water.