As far as I can tell most one-way hashes apply some iterated encryption algorithm to the input data.
What would be the issues with a one-way hash based on some fixed large prime $p$ and a generator $g$ with $h(x) = g^x \bmod p$? Is it just that the methods used for modern hashes are even harder than the discrete logarithm?
It's a fine one-way function, based (as you noted) on the computational difficulty of the discrete logarithm problem. As far as its relation with modern-day cryptographic hash functions, however, it lacks two quite important properties.
Here's an okay definition: a function $H$ is called a cryptographic hash function if it possesses the following three properties:
Note that a general one-way function possesses only property 1. Unfortunately, your $h(x) = g^x \bmod p$ fails property 2. Suppose you had a fixed message $m$ that the adversary knows. Notice that
$$h(m + p - 1) \equiv g^{m + p-1} \equiv g^m \cdot g^{p-1} \equiv g^m \pmod{p}$$
by Fermat's little theorem. By extension, you find that $h(m + k(p-1))$ collides with $h(m)$ for any $k \in \mathbb{Z}^{+}$. So, you can in fact efficiently find infinitely many collisions with any $m_1$, which is relatively exciting, I suppose. And since $h(x)$ doesn't have property 2, it can't possibly have property 3. On the other hand, SHA2 and SHA3 (as two example candidates of modern-day hash functions) are strongly suspected to satisfy even property 3.
Also, even if we wanted a one-way function, modular exponentiation is dreadfully slow. Even with the repeated-squaring method, it's still painful. Some researchers at Intel did a benchmark with a modular exponentiation implementation and found that their implementation took nearly 450,000 cycles (!) for a 512-bit modular exponentiation. On the other hand, the Crypto++ Benchmarks report a figure of 17.7 cycles per byte for SHA-512. Now, I do admit that these were ran on different processors, but the difference is pretty spectacular.
So, in sum, there are essentially two reasons we use hash functions like SHA2: (1) they offer two extra properties, both of which are quite essential for many purposes and (2) modular exponentation is so very painfully slow. I'm sure there are more to be found, but I think these two are the most compelling.
