I am also a dual citizen of the US and an EU country. I routinely depart the US, bound for the EU, without showing my US passport to anyone. Nobody has ever said anything about this.
US law requires US citizens to "bear" a valid US passport when leaving the US. So the difference between my experiences and your plan is that I was in compliance with the law because I had my US passport in my pocket. Had anyone asked, I would have been able to show it. Another thing to keep in mind is that the law provides no penalty (there used to be a penalty, but it was repealed in the 1970s). So even if you leave the US without your passport, you are very unlikely to have any serious problem.
Your experience may depend on the airline, however. When I flew to Africa about a year ago, Delta's check-in kiosk asked my traveling companion, a non-US citizen, to scan her visa. Since I was planning to use my US passport on the other end, and we didn't have a lot of extra time, I just checked in with my US passport. I've never flown on Delta to the EU, or at least not recently, but it does seem like you might encounter some resistance if you try your plan with them.
I don't know whether other US-based airlines will have procedures similar to Delta's. The airlines I have flown with showing my EU passport are mostly EU companies, though I did fly to Curaçao using my EU passport a few months ago, on JetBlue, and nobody asked about my immigration status in the US.