Breast milk serves the only good purpose as a source of nutrition and immunological defense for babies.
In my experience, as a mother, the best way to avoid diaper rash, is to change nappies regularly and give babies and toddlers time without diapers, as frequently as practical. Regular baths, without strong soaps, and the purest and mildest of products. A baby should never be allowed to sit in a soiled diaper, as this will cause reddening of the skin immediately.
A breast fed baby is, supposedly, likely to have fewer allergies and stronger protection from illness. Provided they ingest the milk via the mouth only.
New mothers, first time breast feeding, almost universally, in the developed world, suffer sore nipples. Our breasts are swaddled in bras and protected from the ordinary chaffing that toughens nipples and reduces their sensitivity to a suckling newborn. The good news is, this passes and many women continue to have good breast feeding experiences without the application of breast milk to alleviate sore nipples.
Any substance with sugar and protein left in a moist environment becomes a potential breeding ground for bacteria.
As for infections, red eye, sores, breast feeding aids immune defence, but good hygiene practices and keeping babies from high risk sources of infections is a better way to deal with these problems, than a squirt of breastmilk in the eye. I'd recommend medical advice for any of these.