Games using the id tech engine were often affected by visible texture pop in, and apparently the PS3 version was affected more than the 360 version, but the latter still was noticeably affected. Rage uses id tech 5, but I remember playing BRINK (id tech 4) on PS3 which had no mandatory install (it ran from the disc without installing anything to the HDD upfront), but used the HDD extensively for caching texture data. After I upgraded from the standard 5400 rpm HDD to a 7200 rpm HDD I remember texture pop-in was noticeably reduced.
Xbox 360 emulation on Xbox One or Series isn’t really accurately emulating the hardware, instead it translates the original code to something the One and Series understand.