No, DG customers can shop elsewhere. Everything they sell is available at Walmarts, supermarkets, convenience stores, hardware stores, drug stores, online, etc.
My understanding is that dollar stores tend to pop up particularly where mega-retailers like Walmart are too far for basic supplies. That's obviously not universally true, but for many Americans it's not as simple as going to another store - another store might be a 60 minute round trip that costs $10 in gas that they don't have.