If you look at your description all the methods of selection bar one are nothing like methods for publicly elected officials and as such massively subject to cronyism. This is worth a read: https://en.wikipedia.org/wiki/European_Commission#Appointmen...

The one that isn't is the European Parliament which as we've already established is pretty toothless.

On a slight tangent you'll also notice that none of these bodies are s/elected with any commonality of purpose. Hence why IMO the EU struggles to agree on any difficult issues. Just wait and see how much indecision and infighting will occur over Brexit negotiations.

Why on earth would we give up a successfully evolved system of government for this?