alt Hacker Newshello, it's me the Language Fairy.
Sometimes when people use an expression to convey an idea concisely , the details of the imaginary scenario within the expression are less important than the concept being expressed (just so long as the general shape of that scenario fits the thing being discussed).
To be more particular, the exact time it takes to sharpen an ax and chop down a tree are not important here.
people are so far removed from the ax that they don't realize my point. I'm sure the people Lincoln was talking to had more knowledge and so would get a different picture from the modern reader.