Why is the category of fields seemingly so poorly behaved?

There is a precise sense in which the concept of field is not algebraic like, say, the concept of ring or group or vector space etc.: it is a theorem that any kind of mathematical structure that is defined as having a set of elements and some fixed list of total operations of constant finite arity obeying some fixed list of unconditional equations gives rise to a category with certain nice properties (which I omit for the moment). The usual definition of field has a partially defined operation – inversion – as well as an inequality ($0 \ne 1$), which means the theorem is not applicable; the fact that the category of fields does not have the nice properties of algebraic categories tells us there is actually no way of defining fields so that the theorem applies.

So what does being algebraic buy us, and how do we recognise an algebraic category without thinking about the logical form of the definition? Well, a category is equivalent to a category of algebraic structures if and only if it has all of the following properties:

• It has limits for all small diagrams and colimits for small filtered diagrams.
• There is an object $A$ such that the functor $\mathrm{Hom} (A, -)$ has a left adjoint, is monadic, and preserves colimits for small filtered diagrams.

In fact, it follows that such a category has colimits for small diagrams in general, but this fact is not needed in the theorem. Note that the object $A$ is not unique up to isomorphism; this is essentially the phenomenon of Morita equivalence.

This is a very minor comment on this statement:

The usual definition of field has a partially defined operation – inversion – as well as an inequality ($0 \ne 1$), which means the theorem is not applicable

Actually the issue is not that inversion is partially defined. You can of course extend inversion to a total function $(-)^{-1} : F \to F$ by just declaring that $0^{-1} = 0$. This offends our mathematical sensibilities but it can be done. The issue is that if you do this you still need to state a non-equational axiom, namely that $a \cdot a^{-1} = 1$ for all $\color{red}{a \neq 0}$, and that "$a \neq 0$" part is the real problem.

This $a \neq 0$ requirement prevents fields from being closed under products (by which I don't mean categorical products but cartesian products with inherited operations): as you can see from the fact that $(1, 0)$ and $(0, 1)$ are zero divisors in a product $F_1 \times F_2$ of two fields, the precise issue is that a nonzero element in $F_1 \times F_2$ is not a pair of nonzero elements in $F_1$ and $F_2$. So even if we define $(-)^{-1}$ on $F_1 \times F_2$ pointwise (by either taking the inverse or sending $0$ to $0$), the non-equational axiom above is not satisfied. So, by the easy direction of the HSP theorem fields are not a variety in the sense of universal algebra, as John Coleman says in the comments.

From here we could ask something like: what is the best "equational approximation" to the non-equational field axioms? The simplest one I can think of is that $a \cdot a^{-1} \cdot a = a$, meaning that $a^{-1}$ is a kind of "weak inverse." We could also require that $(a^{-1})^{-1} = a$, which implies $a^{-1} \cdot a \cdot a^{-1} = a^{-1}$, and we could also require $(ab)^{-1} = a^{-1} b^{-1}$. (And we just drop $0 \neq 1$ entirely.) It would be interesting to know whether these, together with the commutative ring axioms, imply all equational axioms satisfied by fields. (Edit: I've asked a question about this here.)

These are perfectly sensible equational axioms and they now hold not only for fields but for all products of fields, if $a^{-1}$ is taken to be the operation which componentwise either inverts or sends $0$ to $0$, as above. They also holds for subrings of products of fields which are closed under weak inverse. The resulting category must in fact have all limits, colimits, etc. now, although the coproduct seems to be strange. I guess there must even be a free such thing on $n$ generators but I don't know what it looks like.

Every ring in this category is von Neumann regular.

Interestingly this category includes all rings satisfying $a^n = a$ for all $a$ and a fixed $n$, where we take $a^{-1} = a^{n-2}$ to be the weak inverse operation; this condition is the subject of a classical theorem of Jacobson, who showed that they are commutative.