Usual mathematical formulation of a 2d (closed) TQFT is as a functor from the category of 2-dim cobordisms between 1-dim manifolds to the category of vector spaces (satisfying various properties.)

For example, a pair of pants (a morphism from $S^1$ to $S^1 \times S^1$) is mapped to a linear map $f:V\to V\otimes V$; similarly a pair of pants in the other direction, a morphism from $S^1 \times S^1$ to $S^1$ is mapped to a linear map $g:V\otimes V\to V$. Then there are axioms (of the symmetric monoidal category) which say that $f$ and $g$ are essentially the same, reflecting the fact that both $f$ and $g$ came from the same pair of pants.

For me (as a quantum field theorist) all this seems very roundabout. The extra axioms are there to ensure what is obvious from the point of view of the two-dimensional field theory; the extra axioms were necessary because the boundaries are arbitrarily grouped into "the source" and "the target" of a morphism, by picking the direction of time inside the 2d surface. (It's called "the Hamiltonian formulation" in physics.)

I think you shouldn't introduce the time direction in the first place, or in the physics terminology, you should just use the "Lagrangian formulation".

In some sense, the idea of "morphism" itself implies an implicit choice of the direction of time. However, you shouldn't introduce the direction of time in an Euclidean quantum field theory. So, you shouldn't use the concept of morphism. The idea of "arrow" itself is so passé, it's a pre-relativity concept which put paramount importance to "time" as something distinct from "space".

So, I would just formulate a 2d TQFT as an association of $f_k:Sym^k V \to K $ to a Riemann surface having $k$ $S^1$ boundaries, and an axiom relating $f_{k}$ and $f_{l}$ to $f_{k+l-2}$.

Why is this not preferred in mathematics? Yes in the physics literature too, the transition from the Hamiltonian framework (pre Feynman) to the Lagrangian framework (post Feynman) took quite a long time...

Or is the higher-category theory (of which I don't know anything) exactly the "Lagrangian formulation" of the TQFT?