There are no real geographic barriers stopping Austin's growth. I am not in Austin but if I have to guess, the public transport sucks balls. You have to have Car unless you live in the urban core. The bigger problem in Austin is, their nimbyism manifested in such a way that City and County constrain the growth by not expanding the infrastructure.
But the land is there - the city can grow into a bigger metro.
Public transit in Austin is laughably bad unless you're on a small set of travel paths, mostly to the university and a narrow part of the city center.
As an example it takes about 20 minutes to drive from my house to the airport, and 3 hours to get there by bus, not counting the walk to the bus stop. The light rail line goes nowhere near the majority of the city.
If we're doing any sort of event downtown or ACL or whatever, I rent a condo nearby instead of dealing with taxis or busses despite only living 15 minutes from the city center and very much "in Austin."
Nobody deals with publix transit if they can afford not to.
As in much of Texas, not very good in most places. If you are central there are some options. And Prop A passed in the last election, which will expand the (single) commuter rail line that exists now (among a few other things). If it is well received, it could potentially expand beyond that. Most residents I know are hopeful but skeptical. Living along Lamar puts you in the best position to take advantage of it, and there is concomitantly a "rapid" bus line that runs regularly North / South fully along it -- I used that on occasion too.
I was fortunate to live in a community on the Commuter rail stop (in Crestview) while I worked downtown and took the train daily for a couple years -- splendid. But though it was packed, I knew nearly nobody else in person that had used it, and many did not even know it existed -- such is the small areas it served.
IMO the best (commuter) features of Austin is it has the room to grow. Unlike Houston (or Dallas, I assume), it has much less roadway infrastructure and things seem slow to build here. So, despite being in and out of this city since college (~20 years ago), I still feel unsure what the future looks like.