Zinc has only been shown in research to deplete copper when zinc gluconate is used. And even then, the participants were taking up to 1g/day for a year before copper depletion occurred. Poliquin brand uses zinc orotate and aspartate, so you won't have these issues.
A good multi will give you enough copper to counterbalance any POSSIBILITY of zinc induced copper depletion - you'll need only about 2mg/day.
I'd say a good 95% of people can't get enough zinc, due to a combo of poor absorption, high heavy metal load, and high exposure to chemical estrogens. In that way, zinc will actually help to prevent prostate/breast cancer - additionally so because it is a potent anti-aromatase.