Misconception in taking Pumping Length of language {a} to be $2$
Informally, the pumping lemma for regular languages state that any sufficient large string may be pumped.

Your language is finite, meaning if you pick a pumping length that is larger than any string in your language, the lemma is trivially true. In your example the pumping length is $2$ but there are no strings in your language of length at least two. So you can pump all strings of length $2$ or longer (there are none!).

If you try and pick a smaller pumping length (the only possibility is $1$) then you will find that you can not pump. $w = a$ is your only string and the only possible decompsition $w=xyz$ is $x=\epsilon$, $y=a$, $z=\epsilon$. You can't however pump that because e.g. $\epsilon$ and $aa$ is not in your language.

Therefore; $2$ is the minimal pumping length of the language $\{a\}$