Why would being the first at the (horizontal) asymptote mean owning a profitable market? In six months, somebody else is at the asymptote as well. If different models are mostly interchangable quality-wise, then the decision will mostly be made on price.

For a market to be both winner-take-all as well as lucrative, you need some kind of a feedback cycle from network effets, economies of scale, or maybe even really brutal lock-in. For example, for operating systems applications provide a network effect -- more apps means an OS will do better, a higher install base for an OS means it'll attract more apps. For social networks it's users.

One could imagine this being the case for LLMs; e.g. that LLMs with a lot of users can improve faster. But if the asymptote has been reached, then by definition there's no more improvement to be had, and none of this matters.