The EU parliament can not be, and is not, the supreme legislative body for the simple reason that the member states are still sovereign, with the ability to withdraw (though there are some legal questions in some member countries about how that would be possible).
It is the same issue facing the US under the Articles of Confederation, where Congress was effectively powerless to do anything without full agreement of the states.
However the EU "worked around" this in part with the Council and Commission functioning as a sort of an "office of the president" (consisting of heads of the member states and the head of the Commission and a separate Council president) and cabinet respectively, with power delegated from the respective governments of the member states rather than from the parliament.
It is a "workaround" for the fact that a federal EU is currently entirely unpalatable to the electorate in most European countries to make the EU governable - ironically because people are worried about giving the EU more power, while the current arrangement is worse in that regard.