The MEP are not part of the government, they are part of the parliament, which is elected by the public. The EU comission (which is the government), is decided upon by the national governments. Only the head of the comission was elected (indirectly) by the people.
Edit: Btw, the head of the comission is more like what the US would think of an actual president. The president, OP talks about is more like a representing nation.