I used to have a 360 and it was a good rig, easy to fix when something went wrong as well. Although depending on how good a computer you have you could always move it to where you were going to set up the console and just get a controller for it. I'm running a mid gen i5 with an RX570 graphics card plugged into a tv and so far that's been more then enough to run pretty much everything coming out that I've tried without issue.
Either way I would check out xbox gamepass and try the months trial as it is only £1 and should give you access to a decent library of stuff to play although I'm not sure how much there would be for the 360.
Game wise for the 360 I'd recommend these having played them. With Skyrim and the Fallout games legendary or GOTY editions will also give you all the DLC content as well:
Skyrim
Fallout 3
Fallout New Vegas
Dead space
Assassins Creed Black Flag (Blackbucket rides again
)
Xcom: Enemy Unknown/Within
Red Dead Redemption
Batman Arhkam Asylum
Aside from Xcom and dead space they are all open world games so you can always just explore rather then follow the story. In Black Flag I'd stay away from the corners of the map until your ship is fully upgraded unless you like getting sunk................a lot.
