All of these things are partially true, but not fully.
Your yard is only rarely representative of the "natural" ecosystem that would otherwise exist. Generalist species may find your yard similar enough to thrive, but specialists (carnivorous plants are a popular example) likely will not even if they're native to the immediate area.
Non-natives can actually be even more efficient than natives at many things, including carbon capture. Prickly pear is a highly invasive example that outcompetes native crops with higher efficiency photosynthesis, huge adaptability to arid environments, and massive carbon capture potential due to its rapid growth rate.
Speaking of prickly pear, it's a native family in much of the western US, but it remains invasive in yards throughout that range because the ecological succession that would naturally control its reproduction takes decades and isn't allowed to occur. Don't plant it.
Your yard is a cultivated environment. Be cognizant of what you're planting and how it interacts. Natives are often a good choice, but there are perfectly valid reasons to plant non-natives or avoid certain natives.