My suggestion is to get to the root of the problem. Why are there tears spilling out of your dogs eyes to begin with?
Entropion and/or hair on the nose/around the eyes growing into or poking the eyes can wick the tears out. This then causes staining. Without the tears there would be no stain.
Distichia are extra hairs on the inner edge of the eyelid ("inside out eyelashes" as I call them lol). They can cause excess tearing.
A blocked tear duct can keep the tears from flowing out through the nose like they
re supposed to, which can make the nose dry and bristly but also means the only way out for the tears is out the corners of the eyes, which again means stains.
There are a bunch of other eye issues that could be underlying conditions, including but not limited to corneal ulcers, allergies, dry eye (ironically), conjunctivitis, and glaucoma.
I would try to find out what is causing the tearing before I worry about getting the tear stains out. The condition itself may or may not need treatment. If you know what the eye issue is and it's not severe, then just deal with the tearing, But it may be possible to make it so there's no excessive tearing, and thus no tear stains, to start with.
As an example, my dog has perfect eyes in the spring, summer and fall. Not a bit of tearing or staining. In the winter, probably from the dry air (and the extra 7 pounds he's put on isn't helping any >.<), he gets a bit of entropion. This means his lids roll in a little bit and the hair wicks the tears out of the eye and into the hair. This makes tear stains. I know that it's seasonal, and I know to keep an eye on it, but it's not something that really warrants treatment at this point. I could try a tear stain remedy if I wanted to, but I figure it'll go away in the spring on its own, so why bother?