There is no clear name for The Disease. Different dermatologists may diagnose different diseases for the same problem. And there is no clear disease name in the literature that can easily be pointed out. The Disease may therefor be a combination of known diseases, or a combination of existing.

Impetigo

https://en.https://en.https://en.wikipedia.org/wiki/Immune_disorderwikipedia.org/wiki/Immune_disorderwikipedia.org/wiki/Impetigo

Tinea barbae

Seborrheic dermatitis

Blepharitis

Demodicosis

Herpes simplex

Rosacea

Folliculitis

Scabies

Why?

Could explain why impetigo (which matches most from a visual perspective), doesn’t heal by itself within a few weeks. Scabie mites could cause impetigo bacteria to be transported and and to be protected in scabies burrows.
Could explain itch.

Why not?

Itch is probably not as heavy as normal scabies infections.
Scabies usually can occur everywhere, except on the face. Although, they may occur on the face of young children, when they are infected.
I haven’t seen any under my facial skin microscope.
Burrows should be visible, although the oiliness of the face may make burrows less visible.