That would also amount to compelled speech.

I disagree. The legal requirement to apply a warning label is a well known, understood and accepted process that is applied to a myriad of hazards to children and adults. As just one example businesses in some states, most notably California are compelled to add warning labels to foods and other products that could cause cancer.

Clients could refuse to show content that does not have headers set.

On other hand servers might choose to lie. After all that is their free speech right.

So maybe you need some third party vetting list. Ofc, that one should be fully liable for any damages misclassification can cause... But someone would step up.

Compelled to disclaim facts is good compelled speech, though.