> It seems as if one ambiguates the other.

Interestingly this mechanism contributed a lot to Brexit in my opinion, whenever there was legislation the politicians wanted to enact but knew would be unpopular with the voters they'd blame the EU and wring their hands saying there was nothing to be done, while in the background gold-plating the legislation far beyond what the EU was even asking for. This also made it easy for populists to point to many absurd examples of the EU being comically bureaucratic, when a lot of the more nonsensical ones were literally domestic additions. When the referendum happened, the consequences were obvious to everyone.

While no Brexiteer myself, I personally would want to see the European Parliament get a lot more powers and influence before I threw my wholehearted support behind rejoining. Half the problem with the EU is that it's seen as not particularly democratic as a place for legislation to originate, and giving the elected organ more power (including the power to propose legislation in its own right, and dismiss legislation proposed by the Commission with much more prejudice) would go a long way to addressing that I think. The Parliament needs to be more, well, parliamentary in how it functions.