Well, there's tons of different methods of seed dispersal for plants, besides just birds eating seed bearing fruit. Other animals, such as deer, coyotes, etc., also eat fruit and carry seeds. But seeds also spread by floating on the wind (think dandelions), catching rides on fur and clothing (think burs), to literally practically bursting forth from their pods and letting fate decide what happens to them. Additionally, some plants, instead of spreading by seeds, like to spread by establishing root colonies and branching out in clusters. Anecdotally, I know the majority of the prolific plants in my yard don't make fruit eaten by birds, so they must be spreading in other ways.
As for native/invasive species, plants grow where they can thrive. That said, like animals, plants compete for space and resources. If someone wants to let their yard go fallow and wants to help prevent against invasive species, there's a lot of things they can do from cautious weeding to planting native plants and seeds that can be gotten anywhere from local seed/plant swaps, nurseries, and even stateagricultural and department of natural resources programs.