That's not true. Most developed countries have some form of privatized system, falling into your 'semi-privatized' definition. It's more cost-effective.
* pay for private health insurance, or
* pay an additional 2% surcharge/tax on your income
This is to encourage people who can afford it to purchase private health insurance, to lessen the load on the publicly funded system where possible.
The reason that private elements exist at all are NOT because they are more cost efficient. They exist just because some people like to have more comfort and are willing to pay for it. Not sharing a room in a hospital doesn't make you healthier (and is certainly not more efficient), it's just nicer.
I'd agree with your parent-comment that most states have actually figured out that public health care is the most efficient option. Private health-care providing necessary treatments is the more expensive option for society.
Germany, on the other hand, has a regional and national market system with compulsive health insurance, much like the ACA attempted. It's not a public only system by any means.
It's not about sharing a room, actually. It's about reducing the burden on the state, providing more efficient service for cost for everyone.
Further, the main issue with public-only systems is wait times. In Canada now the wait times are extremely long - in the area of 20+ weeks. http://www.ctvnews.ca/health/healthcare-wait-times-hit-20-we...
That's not about comfort, it's about getting treatment.
The difference really is in who negotiates. In the US, it is usually individuals organizations or companies - in Germany it is mostly the state, though people still have choice. It's not administrated by the state, though - it's administrated through public/private partnership. Personally, I think it's a great system, and I wish the US would have implemented a version of it more successfully.