Hi Carsten,
Furthermore, Mozilla recommends to use the appropriate header text wherever possible https://www.mozilla.org/en-US/MPL/headers/%C2%A0and mention it in their FAQ https://www.mozilla.org/en-US/MPL/2.0/FAQ/#apply
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. Reuse-tool allows you to use template for specific needs like if you want to go that route https://reuse.readthedocs.io/en/latest/usage.html#templates
Maybe others can provide more details and insights.
Best, Nico