Hello, an Australian here (and an Economics major, though they are a dime a dozen these days so don't put too much stock in that ).
Compared to the current US system, Australia's superannuation is miles ahead - truly you're comparing a BMW to a Corolla. Step back though, and there remain some flaws in the Australian system. Further more, there are some tenants of our system that are fundamental to its' functioning effectively, and I fear they would be stripped out if America tried to implement the system.
Firstly, as others have pointed out, our 'Superannuation' exists alongside a social security (we call it pension or welfare) system. This system has some strengths, and some flaws.
Whilst the 9% (now 9.5%) superannuation has greatly increased our national savings, and over the long term left our budget in a much better position, it is actually insufficient. Over the medium term, we're bumping our compulsory employer contribution up to 12% (and some question whether it should be 15%). For most younger people this will mean they have a very healthy nest egg when they retire (assuming that for us, the retirement age will then be higher). As others have pointed out, one flaw is that you can draw down on your super before you can access a pension, resulting in you being a drain on the public purse eventually, after you've overutilised your super.
An issue with the pension specifically, is that its means-testing *excludes* the family home. So, if you have a home worth $1.2 million, but have no other assets and only $50k in cash, you'll be assessed as having $50k, and deserving of a full pension from the State. In the long term, we'll have to bite the bullet and change this. But politically that is very difficult, as there are voters both in those homes, and in the form of their adult children who want to inherit the prime real estate. One proposal has been a 'reverse mortgage,' whereby the state receives some equity in your home in return for a pension - eventually with the option for your children to buy the state out after your death, or to simply receive the remaining share of a sale.
I have seen people in this thread mention 401k's. The primary difference as I understand it is they are not compulsory. To work as intended, Super really has to be an across-the board requirement. Otherwise some people spend big while younger, then draw a pension in their old age. I support favourable tax treatment of super to a point, but in Australia we're starting to see evidence of substantial loopholes that need to be closed.
One key element of the system that I fear wouldn't survive a transition into America is industry super funds. You DO have the freedom to take your super business almost anywhere you want. You can have it managed by a bank, manage it yourself, or use some sort of independent provider. However most industries have not-for-profit industry super funds that don't pay commissions to analysts. Over the last 10-15 years, these funds have actually OUT-PERFORMED the investment decisions of most commercial providers. It has resulted in our savings being MUCH higher than they might have been if more people were with high-fee, under-performing private providers. I am concerned that in America, the finance lobby would be successful at largely cutting out not-for-profit providers (NFP's are literally the default provider for an enormous number of employers).
All-in-all, the sooner America shifted to a system like this, the better placed the nation and its citizens would be, and the less money the Government would lose - with some caveats.