If some people are more athletic due to good genes, can there be a genetic influence on how much exercise is healthy for you?

Probably. However, since no one knows anything about this, and it's really impossible to study scientifically (since the benefits of exercise are long-term and experiments can only be conducted short-term), take this experienced physician's advice -- find some kind of exercise you enjoy doing, and do it, and if you have someone to join you, count yourself lucky. And thanks for an excellent question.