This is true. However there's a potential way to fix it, and it would only take one member state, no matter how small to start the landslide. And it's basically the same thing that happened to the US Senate.

The Senate, like the Council of Ministers, was intended to represent the member states. Originally, Senators were appointed by state legislatures. A few states started to have direct primaries, with the legislative appointment amounting to ratification of the popular choice. Then this was made the way for every senator by the 17th amendment. (It took until 1913!)

The same thing could happen to the EU council of ministers. It's a little more obfuscated because the representative to the council of ministers is formally a delegate of the government of the EU member state. But, since the process is under the control of the member state, there's nothing stopping them being directly elected by the public of that EU country. They can simply decide that that's who they want to send. Unlike any other democratic reform of the EU, this doesn't require consensus among all member states (which in practice, means the political elites).

There would have to be some kind of groundswell of support for it in several nations, as otherwise the government of bigger nations would do everything to make an individual small nation regret it; unless they would pay some political cost. But if there is a movement to democratise the EU, this is the one move that doesn't need the permission of the incumbents.

because it is a perfect tool for all governments to 'wash' unpopular political bills that would be a political suicide in local politics.

You're asking the group profiting from current arrangement to change it.

That is also true - national countries sabotage any reform.

Basically the EU would have to be designed anew from the get go, but lobbyists will prevent this from ever happening.