I've gone over OPs posts again to try and give some objective advice.
For someone with anxiety, issues such as the riots in France, shootings in America, stabbing etc will be a cause of concern and to some extent, it is entirely justified. He is being a protective parent by not wanting to take his family to a place that has civil unrest.
He becomes quiet and looks at his phone, likely because he actually does feel bad for feeling the way he does. With anxiety comes feelings of guilt and feeling like you are a burden on your family. It's evident he feels bad with things he says but at the same time, he will find it near on impossible to stop these intrusive thoughts. Which in turn, can lead to clinical depression.
OP, why do you think he is exaggerating it? I would suggest reading up on anxiety and doing some research to get an insight into how he is feeling. As they say, with knowledge comes understanding.
Many people don't like to take medication as they feel it is a sign of weakness (which it's not) and there is still stigma around the issue of mental health as demonstrated on this thread. When there is a lack of understanding or empathy, people are unlikely to talk about what is bothering them. Does he lead a normal life aside from the holiday issue? If he does, I can understand him not wanting to go on medication solely for that issue.
He struggles flying with you and the children because he is worrying about you all which actually shows he cares a great deal. His intrusive thoughts have just got a little out of hand on the subject but it does explain why he can do holidays alone with his friends - because he only has himself to worry about which essentially lowers his anxiety.
@MrsBennetsPoorNerves Thank you for your kind words and well wishes. I disagree that he is guilt tripping the OP, he has anxiety about travelling and understanding, support and encouragement is needed.
@Tincan5555 I completely agree with you and not wanting to take medication can also be part of the anxiety so he needs support and love to get his mind in the right space to be able to deal with the matter.
@ImNotReallySpartacus OPs husband is not a 'wuss' for having these intrusive thoughts and I would hope people would be ashamed if their children had that viewpoint.