Whether the sunglasses good for eyes or not shall not judge by the prices but the quality of the sunglasses. There are some cheap sunglasses that made of good quality wit UV blocking lenses. Wearing such sunglasses, it will protect the eyes from intensive sunshine and UV rays. So, they are good for eyes. But if the sunglasses is just for looks and can't block UV rays, even if they are expensive, they are bad to eyes.
Of course. cheap sunglasses also have UV protection. As long as you get your cheap sunglasses from a reputable rertailer, they should have UV protection. And you can check them by yourself to see whether they have UV protection. Generally speaking, there is a label on the sunglasses lenses saying UV 400 or 100% UV protection, which means the sunglasses can give your eyes full protection from harmful UV rays.
I am sorry I could agree with you because to some sunglasses manufacturers, the less investment means the less input to the sunglasses which will results in the poor quality of sunglasses. However, the main function of the sunglasses is to protect you from the sunlight, but if the UV protection is poorly set, how can we be persuaded to buy such sunglasses? Therefore, I really think you deserve other good sunglasses even though a little bit expensive.