Cleaner shrimp and cleaner gobies may eat the parasites from the fish, but they do not eradicate them from the system. It's like a person who likes mushrooms turned loose in the forest... to think the mushrooms would be all gone from the woods is wishful thinking.
The only way to get rid of it in the system is remove and treat all fish, and leave the tank fallow long enough to break the life cycle of the parasite. With no host, they will eventually die off. I can't speak for the effectiveness of UV, but I believe if it is a strong enough sterilizer it can help a lot.
What usually ends up happening IMO is that the system has ich in it forever, but healthy (and eating) fish are able to ward off outbreaks. I do believe that garlic supplements can prompt a fish to eat when it doesn't feel well, and if they don't eat they will die no matter what the root cause.
For example, one of the first fish I ever put in my tank "popped" a little bit of ich. It went away shortly. There have been no visible parasites on any fish since (nearly a year) in my tank, but I promise you they are there.