Mathematicians have sometimes defined TQFTs in the way Yuji suggests. Indeed, Getzler and Kapranov define the notion of "modular operad" for precisely this purpose (it formalizes the relations between $f_k$, $f_l$ and $f_{k+l-2}$, as well as between $f_k$ and $f_{k-2}$). Earlier, Kontsevich and Manin axiomatized Gromov-Witten invariants along these lines (without distinguisning between incoming and outgoing).
Perhaps the main reason that mathematicians use the language of symmetric monoidal categories is that this is very familiar to them. If you want to explain the idea of a TQFT to the average mathematician, it's easier to say "it's a functor" than to say "it's a collection of linear maps $f_k$ satisfying these relations..."
In addition, there are many very basic examples where the distinction between incoming and outgoing is really important. For example, if $A$ is any associative algebra, then the Hochschild cohomology $HH(A)$ of $A$ carries maps $HH(A)^{\otimes n} \to A$$HH(A)^{\otimes n} \to HH(A)$ indexed by Riemann surfaces of genus $0$, with $n$ incoming and one outgoing boundary components. However, $A$ needs to have a great deal of additional structure -- it needs to be a Calabi-Yau algebra -- in order for this to extend to a fully-fledged TQFT.
As for Yuji's last point, I wouldn't think of the higher-categorical formulation of TQFT as a version of the Lagrangian formalism. After all, for $0+1$ dimensional TQFTs, the higher-categorical formulation reduces to the usual Hamiltonian formalism.