Simple proof of $A_n$ being simple?

Question

I come across an incomplete proof of the simplicity of $A_n,n\geq 5$, in a textbook. The proof looks simple, so I was attracted to it. Here is the proof after I filled in the details.

1. Everything in $A_n$ can be generated by $3$-cycles. To see this, just note that $$ (12)(13)=(123),(13)(24)=(123)(234). $$
2. We now show that every non-trivial normal subgroup of $A_n$ contains a $3$-cycle. Let $\alpha\in A_n$ be an element which permutes the least number of elements in $\{1,2,\ldots, n\}$. We show that $\alpha$ is a $3$-cycle.
3. $\alpha$ cannot be the product of two disjoint order 2 cycles since if $\alpha=(12)(34), \beta=(345)$, then $\beta^{-1}\alpha^{-1}\beta\alpha=(345)$, contradicting the minimality of $\alpha$. If $\alpha=(12)(34)(56)$, let $\beta=(123)$. In general, $\alpha$ can not be the product of disjoint order 2 cycles.
4. Now, we know that the only two cases are $$ \alpha=(1234...)\text{ or } \alpha(123)(45...) $$ the former one must permute at least $5$ elements; the later must permute at least $6$ elements. However, in the former case let $\beta=(234)$ and in the latter case let $\beta=(132)$. Calculating $\beta^{-1}\alpha^{-1}\beta\alpha$ in both cases gives $(143)$ and $(15324)$, which leads to a contradiction, because they permute less elements than $\alpha$.
5. Finally, note that any two 3-cycles are conjugate to each other. So any normal subgroup is the entire $A_n$.

My question is: although this proof is significantly shorter than the other ones I see, I find it quite difficult to understand how those permutations (such as $\beta=(132)$ in the end) are chosen.

1. Why are we interested in the commutator $\beta^{-1}\alpha^{-1}\beta\alpha$? It makes sense to consider $\beta^{-1}\alpha\beta$ since it is related to the definition of normal subgroups. So, why we consider $\beta^{-1}\alpha^{-1}\beta\alpha$ instead?
2. How can we choose an appropriate $\beta$ when we cannot remember the details of this proof? For example, why choose $\beta=(132)$ and $\beta=(234)$ at last? Do we really need to try all possible $\beta$ one by one?