Which of the following has primary responsibility for eliminating alcohol from the bloodstream

There are three ways of eliminating alcohol from the bloodstream: through the kidney, lung as liver.

The primary function of the kidney is to regulate the acidity and basicity of your bloodstream. Alcohol is one of them because alcohols are basic. They have to balance out the concentrations of ions in your blood. The excess waste are excreted as urine. 

The lungs also eliminates alcohol when breathing out. As alcohol easily vaporized into gas, it can be eliminated through airways.

The liver eliminates alcohol through chemical breakdown from alcohol to acetic acid with the aid of enzymes like alcohol dehydrogenase. The acetic acid is then used to produce fatty acids in the body.