I finally found an answer for my question. In short, the effect does not arise when one sets the font of each control individually instead of setting the font of the containing form. This way, the auto scaling feature works as as it should. Interestingly, setting the font of the controls changes the auto-scaling behaviour even if the AutoScaleMode property is set to AutoScaleMode.Dpi, not only when it is set to AutoScaleMode.Font.
As a pragmatic solution, we created a small command line program, which reads the designer.cs files, scans if all controls have an explicit font assignment, and if not, adds the assignment into a newly created copy of the designer code. We embedded this program into our automatic test suite, so whenever a form gets new controls, or a new form is added, and the dev forgets to add the explicit font assignment, the tests will fail. In between, we have this solution working from the time I asked this question first (4 years ago), and it saved us from scaling problems for several times since then.