I certainly didn't vote for Mr J.
I certainly didn't vote for Mr Cameron, but that doesn't mean there was no democratic involvement in his becoming PM. Juncker is the president because the EPP won the most votes in the EU parliament and they has chosen him as their representative*.
The council is composed of commissioners who were chosen by directly by each of the democratically elected national governments, and the whole council is then approved by the democratically elected EU parliament.
The EU could be more directly democratic but it can only do so by removing control and sovereignty from the various democratically elected national governments. As it stands, it's a compromise between sovereignty and direct democratic accountability; and it cannot become more democratic without the national governments surrendering more sovereignty. As it stands, it's hugely more democratic than any other international organisation and, were we to leave, we would find ourselves involved in many of the same negotiations but with far less democratic oversight.
* - De jure the parliament has input into the choice of president, de facto they choose him. It was widely understood across the rest of Europe that Juncker would be president if the EPP won the most seats before the EU election.