Do you realize that the current convoluted lack of accountability is largely a result of the
lack of deeper integration?
E.g. the EU is limited in how much of the power that is vested in EU organizations that can be transferred to the parliament, because much of the power that is elsewhere is there as a side effect of that power nationally being delegated to the sitting cabinets, and so in most countries it will take constitutional changes to hand that power away from the cabinets, that exercise it as part of the EU Council and EU Commission, to the parliament.
This is a "workaround" to problems similar to what the US endured under the Articles of Confederation (where the central government was pretty much powerless to implement many decisions because the states simply could decide to not follow decisions they didn't agree with), and a lot of the work on deeper integration in the EU has focused on how to change this situation to grant the EU parliament more of these powers.
There are plenty of problems with the approach. But there are also plenty of "Europhile's" for whom reforms are simultaneously about tighter integration and democratizing the decision making, and in fact it's hard to find anyone that are happy with the current power split between the EU Commission, Council and Parliament.