sebrohic dermatitis is NOT just a scalp issue you can get seb derm anywhere it mostly effects the facial areas scalp, ears , nose, men get it a lot in the beard areas and moustache some people get seb derm on the chest and the most common symptom is continually flaky scaling crusts. so i don't know were your getting your research but seb derm is far more broad then just a scalp issue.
also its quite possible ec and many diseases originate from a pathogen but i would say the issue lies in the pathogens which live in our gut rather then just on our lips. maybe you should look up fmt(white shark can direct you with some info)
in regards to eczema look into dupilumab which is coming out in the market in 1-3 years it practically cures eczema and clears the skin within 2-4 weeks. and no its not a anti-bacterial or anti-fungal or anti-viral drug is a simple biological drug that switches a main cell off in the process on inflammation.
i think gut issue is massive and maybe there is a imbalance which has occurred and thrown the system out of balance who knows.
i just can't understand why people don't at least consider ec as a form of eczema/dermatitis
each to their own
good luck with your idea please let us know if you kill this pathogen!