A parliament that cannot propose legislation is a parliament in name only. It's a check/balance, I'll give you that, but it's not where the power lies if it cannot propose and effect a change that it wants to.
In the UK, you elect an MP. That MP directly votes on, and can propose legislation. The "other" house, the House of Lords, can only delay any legislation that the House of Commons votes for by returning it with recommendations a maximum of 3 times. After the third time, if the House of Commons again votes it through, it becomes law (subject to Liz' royal assent, but that's not being withheld...).
This is effectively the inverse of the European "parliament". The EU commission decides what laws will be proposed, the parliament (the people who the people elected) then get to horse-trade the deal until the parliament and the commission agree, and then all countries must adopt the law. This is a significant reduction in the power of the people.
As a bonus, the commission are basically immunised against any effects of their political machinations, the only way for a member of the EU commission to be removed is if the parliament unanimously votes to remove all members of the commission at the same time. Yeah... Not gonna happen.
So to summarise: you have an un-sackable body that is the only group who can propose legislation, which gives them the ability to apply enormous pressure to the elected representatives (oh, you want X do you ? Well make sure you vote for our Y and Z and then we'll consider it). And then everyone is forced to accept the results of this as law.
Sorry. That sucks. Given the mission statement of ever closer union, the desire to raise an army etc., and the binding nature of EU law as supreme, the mismatch in democratic power within the EU *should* be concerning IMHO. Whether it's sufficiently concerning to brexit is a different argument, but I think it certainly played its part.