Actually they offer exactly like Medicare and SS. They just don't do a good job of it, which is why the government programs as the difference between abject poverty and self sufficiency for most of America's elderly.
Private companies don't offer the same thing as Medicare. The reason is that the government subsidizes 80-90% of the Medicare premium.
If a private company offered complete coverage to seniors, it'd cost, let's say $500 per month (it'd be more, I'm just picking a nice easy-to-chop-up number). The government provides $400 of those benefits for "free" (you know what I mean - the recipient paid for it all their life, but now pays nothing), leaving the recipient to cover the other $100 of their medical needs.
They can do this by either literally picking up the hospital bill for 20% of their claims, or some people buy Medicare Supplement plans, paying a monthly premum to cover anything the government doesn't.
The other alternative, & the point I'm getting to, is Medicare Advantage, where the government says the private company has to offer the exact same coverage as Medicare but then the government will pay the insurance company $390. The private company figures it'll only cost them $380 to do so, meaning the recipient gets the same $400 in benefits from before, the government saves $10, & the private company makes $10 in profit.
The recipient still needs to cover the extra $100, so they can buy a more expensive Medicare Advantage plan (you can offer better benefits than Medicare, but not worse) or pay the difference at the hospital, same as above.
All that was a long way of illustrating that private companies *can* offer the same benefits as the government & do it cheaper. The fact that 8.4 million seniors are enrolled in Medicare Advantage, the goverment wants to continue it, & the private companies want to continue it, means it's a win-win-win.
As this headline from the US Government Accountability Office says,
Enrollment Increased from 2010 to 2011 while Premiums Decreased and Benefit Packages Were Stable.