As long as you classify as annoyances the overheat deaths of many poorer and/or frailer (and a few wealthy-but-unlucky) members of your developed world, then indeed you can call this annoying. Also you can drop the "will be" and start using "is".
Basically, the spherical cow mindset of "just police the borders harder" is wishful thinking. The world is interconnected in multiple aspects, not just physical borders. Even if you succeed in immunising yourself from all side effects, your immediate neighbor may not be successful, and the instability can come for you through many mechanisms. Prices of food, breakup of the EU, wars, authoritarianism in your own society. Just look at Canada dealing with tariffs through little fault of their own. It isn't as easy as declaring yourself as immune from international instability.
Without trade interdependencies wars seem much more attractive to powerhungry leaders of countries, engineers can probably earn a living building killer ai robots.
Thats one of those things that, once it gets bad enough, it will be solved rapidly with automated sentry turrets or similar technology and will turn into an annoying logistics problem of keeping the sentry turrets armed. It isn't correct to point at that as the thing thatcmakes the original statement untrue.