Most large companies stay out of a 10-foot-pole range of anything that legal labels with the L word (liability). The fact that not giving reasons because of liability is commonly accepted wisdom in recruiting means that deviating from it takes a lot of work.
Specifically, a talent acquisition manager/executive would have to make a business case about how exposing the company to liability from outsiders would somehow pay for itself.
The only plausible argument that I can think of is that you could market your recruiting process as transparent and then get some additional candidates based on that. But.. can you really postulate that some candidate that is reading about your recruiting process in depth would be highly enticed by this to actually apply, as opposed to verbiage about other perks? Can you guarantee that having a free "recruiter feedback on your application" service available to anyone out there won't create a great incentive for random candidates to apply just for that? What if there is a data leak and all this prejudicial, personally identifiable information ends up on the web?
That's just what came to mind in 5 seconds. If the argument for creafting, storing and providing the rejection information sounds like a stretch, it's because it's a tough sell.
About the actual legal requirement, OFCCP compliant companies have to record disposition reasons for candidates that they don't hire when some conditions are met (candidates have expressed interest in a position and they were evaluated based on "substantial information"). In my experience, these disposition reasons try deliberately to be vague. Regular companies don't, and most likely won't because of the liability issues mentioned above. For reference, here's the go-to FAQ on OFCCP requirements: http://www.dol.gov/ofccp/regs/compliance/faqs/offaqs.htm