The reason it works for three (but not other numbers) is the uniqueness of trilinear functionals.  Let $G = \mathrm{GL}_2(F)$, $F$ a local field and let $\pi_i$ ($1\leq i\leq 3$) be three irreducible admissible representations.  Then there is at most one $G$-invariant functional on $\pi_1 \otimes \pi_2 \otimes \pi_3$ (this is mostly due to D. Prasad).  Now let $\pi_i$ instead be global automorphic representations, and $G$ the adelic group. Then one such functional is given by the integral on the LHS of Watson's formula, so by the uniqueness it must factor as a product of local functionals, and this "explains" why the integral is Eulerian.

For the general question of when one should expect such integral representations for L-functions see the recent work of Sakellaridis and Sakellaridis-Venkatesh.