gotta be bs. full games do not work without a jtag or rgh, period.
There was a glorious couple of weeks (for everybody else -- I had a JTAG) where shared profile was a thing and you could download a profile and associated games and DLC on said profile (some whined about not getting trophments on their main profiles) and some of those were pretty feature complete for the time with most of the big games and associated DLC represented. Long since blocked but theoretically there could be something out there with it installed (I don't know if the specific profiles used were later blacklisted in updates but it would be a solid bet).
There was a marginal exploit called LIVE2CON where the CON files (uses for saves and whatnot, and the same underlying format as LIVE aka XBLA and DLC, also PIRS aka what various game of the year* editions used to install DLC to a drive without it. STFS if you want a search term) where some games would take the LIVE files redone as CON files and load them accordingly, mostly only seen for some guitar hero DLC I think but I never investigated this one fully.
There were a handful of games that various glitches saw able to bypass the end of demo restrictions and enjoy a full game. You will have to go hunting for those as obviously they got yanked from the store and I don't know of those versions were blacklisted on newer kernels but again solid bet.
I don't think we ever saw anything like the disc swap tricks some COD hackers used for this but it might have been a theoretical exploit.
I don't know what is available for the original 360s with memory card slots that work with the XBLA games some things had bundled on, likewise some larger NAND models were made that bundled things with them as well.
*that is also a thing you can use, even more so these days where you don't have to worry about getting access to the hard drive and instead have USB. The things available in PIRS or related concepts is very small though as most GOTY DLC was either on disc or a nice one time use code. That said technically serious sam which was sold as disc and XBLA of sorts had a PIRS version you could inject. I don't have a complete list to hand and there are some asterisks for things like I think was fable that worked far better with the install disc rather than injection.
If being an absolute pedant as well then I would note that flashed drives could run the various XBLA games collected and bundled on disc (a surprising amount more than many realise but also far from the best and brightest of the service -- you are not going to be sitting there playing perfect dark, bomberman XBLA or the other cream of the crop titles)