> Given that EU wages have increased much more than US wages we can conclude that unemployment has very little to do with wage increases.
No, we cannot. Unemployment and wage increases could be perfectly positively correlated and still be completely consistent with EU wages rising faster in the US when the EU unemployment rate is twice the US rate.
All we can conclude is that the EU unemployment rate to wage increase function is not the same as the US function, so that a given level of unemployment does not result in the same rate of wage increases in the EU and the US.
There are huge social safety net differences between the EU and the US that might affect how urgent it is for an unemployed person to find a job, and so could affect what employers have to offer to get workers.
In the US, being out of work is a major disaster for many people. If you are in a state that has not expanded Medicaid, you may not be able to afford even the cheapest health insurance for more than a short time with no money coming in if, like many Americans, you do not have much in savings.