what you're neglecting to consider is what would have happened if someone moved to Austin (say a wealthy person from SF) and that expensive new apartment didn't exist.

The mechanism by which new construction drives down rents is that people that need a new apartment are in less competition with existing residents in older worse apartment buildings.

So the newcomer from SF moves into the expensive new apartment, which means that there's less competition for decades old apartments, which means that when one of those is vacated there is less price appreciation on that product.

If there is a scarcity of apartments what happens is that when a decades old apartment is vacant it is filled by a wealthy newcomer and the landlord increases the rent accordingly.