Well that is the problem: There is no good way for people to organise effectively on an European level, or at least it is much harder. By comparison is very easy for a determined few to organise at this level, which is why you have tons and tons of lobbyists in Brussels. You can partially observe this when "Wutbürger" protest against local changes like building a new railway station (months of protest, sitins, general disorder), but you barely get two days worth of protests against the ECB on the occasion of the opening of their new head quarters in Frankfurt.

Note it was not the ECJ which struck down the law first, they did it after the BGB did strike it down, partially because they did not want to get into a fight about who has primacy in decision making I believe (the BGB still claims to have the last word, changing this would be unconstitutional in Germany).

I don't think politicians "hiding behind the EU" is what is actually happening, what actually happens is collusion to shift away power from the people by obfuscation and misdirection. People like Schäuble have active contempt for democracy (According to Varoufakis he told him "Elections can't change economic policy"). See for example this discussion of ECB policies (https://www.yanisvaroufakis.eu/2017/03/14/19258/).

I truly believe the EU has to go, in order for Europe to not further slide into a corporatist neo-liberal nightmare. It is very unfortunate that this push is lead by right-wing nationalists.