We are just back and we did have some issue with the 144-hour visa-free entry (important: don't call it a visa, it is a visa-free entry - two different things legally) at my little airport here in Germany. The issue is that it does not depend on the airline, but ultimately on the individual agent you end up with. Ours had never heard about this, but then called her supervisor and he knew about it. It still took them 15 minutes to locate it in their computer system. Once they had found it everything was ok. I kept telling her that she had to look it up as a transit, but she kept insisting that it would only be a transit if we had a boarding pass for our onward flight. Which of course we did not have. So, my advise would be to be there early, be insistent that such a thing exists, that it will be in their system and that they should escalate it to a supervisor. Then be patient. And ultimately if they make you miss your flight, they will have to pay you 600 € under the EU air travel regulation. Good luck and have a great trip!