I actually took a course on modern urban economics in grad school and this topic came up very early: the consensus is pretty much that there is no consensus (go figure).

One thing that you should keep in mind is that the evidence for this phenomenon is highly anecdotal. While there are many prominent examples of cities with their most affluent neighborhoods in the westerly part of town (Los Angeles, London, Paris) there also are many examples which show quite the opposite: Manhattan's most famous well-heeled neighborhood, the Upper East Side, is a prominent example. Another example is Miami, where the affluent Miami Beach (technically a separate municipality, but you get the point) is to the east of "Downtown," and affluent suburbs like Coral Gables are also to the South of the city. Cities outside the United States also exhibit examples which run contrary to this "Common Knowledge," off the top of my head: in Santiago (in Chile), Bogotà (Colombia), Madrid (Spain) all have their most expensive residential real estate in the eastern side of the city (the fact that these are all cities in Spanish-speaking countries is purely a coincidence).

Each city has unique factors which will drive neighborhoods to be more or less desirable, and these factors can change over time. In the aforementioned Manhattan, for example, a trend which started in the late 90s saw the neighborhood of the Greenwich Village, technically on the west side of the island, exhibit even higher real estate prices than the storied Upper East Side. In the Italian city of Milan (for I cannot answer a question without looping it back to my real area of expertise) the most desirable neighborhoods have flipped from east to west as the city developed over the centuries: The west side seems to have been generally more desirable until a large park was built on the east side of the city in the 19th century, quickly making the that side more desirable for the city's wealthy residents. The east side remained the city’s most expensive residential neighborhood through the city’s period of industrial growth and through the two world wars, until post-Second World War housing policy made new developments far to the west of the city's center the most desirable addresses.

There is an actual urbanist here (u/MrDowntown) whose answers I scanned to see if they had anything which might contradict me, but I don't think they do. You might be interested in this short answer of theirs from awhile back in any case.