Why are theoretical hash constructions based on the hardness of the discrete logarithm problem not really used in practice?
I would say it is largely due to


  the fact that discrete log functionalities tend to be computationally expensive and harder to implement on specific hardware (compared to constructions like SHA-2)


In many (perhaps most) use cases, hash functions are not primarily used for their cryptographic security properties, but for efficient and comfortable data management, e.g. in databases. So complexity of running the algorithm should be minimal.

In addition, good hash functions do provide strong security and have several advantages over applications of finite group schemes (where the discrete logarithm applies). In particular, they provide extremely little algebraic structure, meaning there is not much to attack in the first place. This is also the reason why many consider hash functions to be quantum computer resistant, while it is already known that Shor's algorithm can solve the DLP efficiently.

In short, I do not know of any advantage that a DLP-based hash function would bring about, while there certainly would be some shortcomings by comparison.