I don’t think I would put my dog on antibiotics just to get rid of tear stains. But if you use baby wipes on the area and then a blob of vaseline, that will help because the vaseline is pretty waterproof and the tears run off and dissipate. Not a cure, but it helps.
There was an old trick used by show dog handlers, they made a paste of equal parts of Fullers Earth and Boric Acid Powder mixed with water. Put it on the stained hair, don’t get it into the eye, every day for a couple of days, that bleaches it out and then use only when necessary.
I never thought it a medical problem that a dogs eyes tear. Usually on longhaired dogs it is because the hair is irritating the eye, (also could be a medical condition where the eyelashes are turned inward, but that is pretty severe and the eye does not just tear but becomes highly inflamed). Anyhow, either keep the hair up in a rubberband or consistently keep it trimmed short. If you start trimming you must keep up with it or the hair bristles as they grow out are constantly irritating the eyeball.
My 12 year old akita has suddenly started having runny eyes, the vet says is an allergy brought on by diminishing immune system plus the fact that we live in an extremely high pollen and smoke area. I just use baby wipes on him. And occasionally drops in his eyes if they are red and he paws at them.