Hi Carsten,
I understand your concern and honestly I don't know the right answer or oversee all risks of not using the header. Lawyers might have a interesting argument on this ;)
Personally I'm not so concerned as there remain plenty of hints about the license in the file. There are even projects that only add a LICENSE file in the project root without adding anything on the file.
Still, adding Exhibit A header to each file is probably the safer choice.
Maybe others can provide more details and insights.
Best,
Nico