There are many invasives, which are everywhere invasive, and there are obviously invasive native plants, which are ok in their native area and there are plants which are invasive in tropics, but not so in the med climate. I think it is essential to know, what plants are invasive in your area, and avoid them. Some people say, that they have nothing invasive in their garden, but that is not the point. The point is, that these plants selfseed, the seeds are carried out by animals, or the prunings get in the nature. There is a reason, if scientists tell us to avoid some plants.
But about the lawn substitutes. To put a piece of lawn is easy, but to dig it out is hard work... I have left a bit of Cynodon in our garden, as it was there before, is hard to dig out, is quite easy to control and it is nice to walk on barefoot, no bees or other biting insects in it. If you mow it even once in 3 weeks, it won't flower and the rhizomes are easy to rip off or cut on the sides. Cynodon is forbidden now in Canary Islands, but in Spain mainland it stops growing completely in winter, therefore should not get out of control.