This shows how narrow the responsibilities of the EU actually are.
For example, France lost a key advantage ("competition rules") with nuclear energy essentially because it was considered unfair to other countries. But energetic independence is fundamental in part to economical independence: it's a key aspect of sovereignty.
Another example would be the Euro ("monetary policy"):
> Give me control over a nation’s currency, and I care not who makes its laws.
Or, the fact that external laws (from the UE) can be applied to member countries without approval from the people is IMO another rather clear form of loss of sovereignty: a considerable amount (~20% for France IIRC) of legislation is imposed by external, un-elected bodies.
As it plays on the democratic oversight perception as well, we could also look at the perception/quality of the locally elected bodies, and their general attitude towards the UE: a great example is [2] ("let's ask the people but do it anyway").
That's to say, I believe there are good reasons for people to perceive EU laws as imposed without democratic oversight. It's not 100% true, but overall, it's not an unfair qualification either.
[0]: https://www.europarl.europa.eu/infographic/meps-seats/index_...
[1]: https://fr.wikipedia.org/wiki/Liste_des_d%C3%A9put%C3%A9s_eu...
[2]: https://en.wikipedia.org/wiki/2005_French_European_Constitut...
Disclaimer: I am neither American, nor very educated about what I just wrote.
Also in the last 100 years via the Supreme Court's primacy.
1. Can your country introduce any law without asking the EU for approval? 2. Can the EU mandate your country to implement a specific law? 3. Do you still believe your country is sovereign?
2. Yes, within the areas where the EU has exclusive responsibility and to some extent when it shares responsibility with member states.
3. Yes. Any country is free to leave the EU at any time. See Britain.