This is definitely not true in Germany. And for glass bottle recycling, most bottles are washed and reused up to 50 times before being melted down.
Of course, most of the studies on this are sponsored by the paper industry, and it's really difficult to get unbiased information on the subject.
The really big environmental issue that is not being tackled is anthropomorphic climate change. Really it is not going to matter how many plastic bottles we recycle if we push CO2 over 1000 ppm and which we are on track to do.
[1] https://en.wikipedia.org/wiki/Beverage_can#Fabrication_proce...
[2] http://www.alcoa.com/recycling/en/info_page/why_recycle.asp
Your wikipedia article says:
> When recycled without other metals being mixed in, the can–lid combination is perfect for producing new stock for the main part of the can—the loss of magnesium during melting is made up for by the high magnesium content of the lid.
[1] https://www.alcoa.com/mill_products/catalog/pdf/China_Specs/...
Personally I place high value on raw material recycling. Even if it's more expensive monetarily, so long as it uses less resources it seems worth it to me.
Aluminum cans are currently recycled more than any other beverage container in the U.S, which is good for business and the environment, says the Aluminum Association, because making a can from recycled aluminum saves not only aluminum but 92 percent of the energy required to make a new can