The gentle tinkling alerts people you are there without you consciously doing so.
I think that gentle tinkling might well drive me to distraction. I can't stand rattles or squeeks on my bicycles and if any develop I sort them out pronto. I even serviced my friends bicycle for him, not so much as a favour to him, but because the general cacophony it was making when we rode together was getting on my nerves.
I don't have a bell or horn and usually just slow and say "excuse me" or "on your left/right". Bells and horns often seem to panick people and they are as likely to step into your path as out of it.