(Disclaimer: I've lived in Austin for 16 years, and spent a lot of time here for four years before that while my brother was in school here, which gives me a thimbleful of street cred as a longtime resident compared to those who were here in the 70's, 80's, and 90's. But its long enough that I have seen a ton of change in the last decade in a half.)
There's a couple of threads of thought in your comment that are worth addressing separately.
> our reflex is to say that the markets are obviously failing and that policy should step in in order to preserve the integrity of the culture
The market is only good at doing one thing: finding local maxima to a profit curve. If whatever you're interested in can't be distilled to a profit metric somehow, the market is completely oblivious to it, and there are many intangibles we find valuable that fall into this category. When we're lucky, we have tools to link the thing to economic metrics. Other times we don't have good ways of doing so. An example of the former would be a carbon tax, which front-loads the full cost of introducing carbon to its initial purchase. You alluded to good examples of the latter in the form of cultural preservation, which is not as one dimensional as "gentrification" and includes all kinds of things like dying languages, musical heritage, oral tradition, and yes, places where Willie Nelson and Janis Joplin played early in their careers.
The reason I find this noteworthy is that you're positing the assumption that markets should protect culture and find it puzzling that people think this way. While I think there are people who do so, I think the opposite: most of us recognize that this neoliberal assumption is false and markets do NOT protect cultural heritage. Because for most of us cultural heritage is worth preserving, it needs non-market-driven measures to preserve it in some form.
The second thread of thought I have about your post is that culture is also fluid and necessarily must change over time, and those who seek to preserve the status quo indefinitely are fighting a misguided battle. Obviously this is at odds with my point above but fits in nicely with your emergent qualities point. However, I think you're putting a rosy shine on the practical details of how this plays out in the day to day lives of people actually participating in the market.
I think its best summed up by saying you're not entirely wrong: attempts to limit "market expansion" as a means of preserving some aspects of cultural heritage are a statement of values. However, refusal to do so is also a statement of values. In the latter case, you're saying that unrestrained trade is a thing to hold valuable in and of itself, which is a notion that I think many people are coming to see not as a virtue but a vice which needs reasonable constraints. Does that mean the city should subsidize money-losing businesses just because they're part of a cultural legacy in this town? Probably not, at least not with wild abandon. However, since we have no apparent problem with subsidizing big businesses through tax incentives, I don't think its out of the question to find ways of doing the same with small businesses that provide some cultural conscience to our town.