I had 2 of these trees. Yup, had. I took them down for good reason, too so will share my experience. Sorry to rain on your parade. I do hope it helps.
Cracking: The trees have an upright habit resulting in narrow crotches. This lead to cracks and since this is an upright tree the cracks occured well above anyone's head making them difficult to deal with. Arborists tell me this is not uncommon in these trees.
Reseeding: Acacia stenophylla seed production is huge and from my experience a zillion percent of them are viable.
Conside those graceful strings of pearls hanging in quantity from a mature tree, these are all seeds...
They also will sprout from lateral roots at a distance from the trunk of the tree. I'm not sure if these sprouts can become actual trees but they are unsightly.
This occurred both before I removed the trees and after removing the trees. Yup, the roots are still actively producing sprouts. The roots of 2 trees that are not even here anymore are still using soil resources including water. So even though these trees are no longer there they are using soil resources that would otherwise be available to other plants.
Trash: I love mulch but this tree drops tons of stuff all year. The leaves are leathery and donÂt compost easily. And the seeds that are in that mulch make it more useless.
Experiences of friends: A friend who takes care of trees in city parks tells me that Shoestring acacias drop stuff constantly and that this leaves stains on pavement. Another friend also had them spouting from lateral roots in his yard.
Lousy near pools: The leaves are very long and narrow and leathery and grow even more so when wet making them a real hazard for pool pumps. This tree is on some pool tree lists! Be careful of those lists.
Moral of the story for me has been: Trees in parks and parking lots are in very different circumstances than our yards and should not be relied on entirely for examples.
The lists of recommended plants have probably not been improved since written so consider them with caution.

Wow, thanks for the information. I had no idea. The last thing I need in the garden is something that wants to be invasive. And suckers too. I recently saw it while at a garden tour and it looked so nice. It was a small tree but still. I am glad you told me before I made the mistake. Reminds me of the time I planted running bamboo (in Florida) and ended up digging trenches all over my yard to get it out. Never want to experience any thing like that again.
Thanks
Trini1