Yes and no. There is some truth that certain aspects of modern beekeeping practices affect mites, for example we actually use foundation that encourages bees to produce cells (and therefore bees) a little larger than they would in nature, this extra space around the brood leaves more room for a mite to invade the cell. Similarly there is some evidence screened bottom boards like you describe can help by letting mites that fall off bees fall out of the hive. This isn't a more natural hive design however. In nature bees generally nest in holes in tree trunks, which definitely don't have open bottoms.
There is no magic hive design that solves the mite problem. The big issue is that our bees (in the USA) are from Europe, while varroa is from Asia, meaning our bees have never evolved defenses against them. This is why mites have essentially wiped out feral beehives.
The best hope for long term control is breeding programs breeding for bees that have behaviors that keep mites under control. In the meantime there are a number of things beekeepers can do to help the situation, but at this time they're aren't any known practices that will eliminate the need for treatment.
There is no magic hive design that solves the mite problem. The big issue is that our bees (in the USA) are from Europe, while varroa is from Asia, meaning our bees have never evolved defenses against them. This is why mites have essentially wiped out feral beehives.
The best hope for long term control is breeding programs breeding for bees that have behaviors that keep mites under control. In the meantime there are a number of things beekeepers can do to help the situation, but at this time they're aren't any known practices that will eliminate the need for treatment.