My philosophy is similar to that of Mr. Lofthouse. I feel, as long as it's not from outer space, I don't really care what part of the world it came from. If it can grow & thrive in my environment, then I let it do so (at least until I want to have something else in that spot). Since I consider myself the steward of the land, I will take responsibility for restoring the balance if something gets too far out of hand.
Regarding mimosa- I would not worry too much about it taking over. While it does put out a million seeds, less than half probably germinate; and of those that do sprout, maybe 5% survive the first 2 years without help from a human. The pasture next to me started out with one or two trees about 10-15 years ago. Now, I think there's about 5 trees out of all those years of seeds. Additionally, they're pioneer/nurse trees for things like oak & pecan, and die quickly once the other trees reach the same height & start blocking the light.
The thing that makes them "weedy" is they're hard to kill when they are in the right environment after they're 4-5 years old. Since they readily produce suckers when chopped (great for
permaculture), they either have to have a yucky chemical poison applied, or have to be pulled out by the roots to kill them.
But, no, I've never seen them totally dominate an area when left unchecked. Heck, I can't even get them to create a grove by helping protect them.