If you're going to make the hash browns, there's no chance you're going to be successful with three ingredients. You're missing two keys: oil and salt. Oil in this case is an ingredient, because the potatoes will soak a lot of it up, so you want something that tastes good. I always make hashbrowns (which I do probably 3-4 times per week) with good quality olive oil. It's a bit expensive, but the difference isn't subtle. Also, no matter how salty the cheese is, it's no substitute for actual salt.
Do you boil your potatoes before shredding? I've had good results with boiled potatoes that were refrigerated over night and this way I used less butter to make them. Though I love hashbrowns, I am not yet a master, so I'm interested to get more 'brown knowledge.
I neither boil not shred. I use a very small dice, just made easier by the fact I bought a set of Shun knives last week. I've never been so excited by cooking equipment. The key is to use your spatula to make sure the potatoes don't clump while cooking (med high heat), and to add oil occasionally so there's always some in the pan. Gets absorbed quickly.