Bpeezer I really want to offer my opinions to your bullet question but I would like some additional information. What type rifle and ammunition are you using? And you say that we can’t blame the rifle or bullets for groupings that "are never all that good", so why are you blaming the bullet weight?
As for brass, when reloading try to at least stay with the same brand. Using the same lot of brass is the best way to go. Work up a load with that brand/lot and then stick with it for both accuracy and safety sake. Different brands of brass may well have the same outside dimensions but the inside dimensions well not be the same and that changes how much powder is safe to use in it. As hk33 states, Lapua is one of the better brands, as is Nosler and Norma, but they are expensive compared to Winchester and Remington. Winchester and Remington take more work on your part, separating by weight and deburring flash holes, but the cost savings is sometimes worth it.
I do not have any experience with the Lee Hand Press.