I know that any permutation can be written as a product of transpositions. Does that complete the proof? I don't think so, because suppose we need $2\rightarrow 3$, we can't write that in the form $(1n)$.
As you've correctly stated, any permutation can be expressed as a product of transpositions. Then, as long as we can generate any transposition, $(kl)$ from a product of the form, $(12)(13)...(1n)$, we can write any permutation as some product of $(12)(13)...(1n)$. Consider, $$(kl)=(1k)(1l)(1k),$$ which is indeed a product of transpositions of the form, $(1n)$.