Why is the order of operations in the order that it is?

It's because of polynomials. Polynomials want to be written (5 * (x³ )) + (3 * (x² )) + (4 * x) + 1.

Notice that this has a lot, a lot, of parentheses. Around the time that this notation was getting formalized, polynomials and their properties were considered the most important use of algebra. Over time people got lazy, and stopped writing the parentheses at all, but everyone knew what was meant. This eventually became formalized as an "order of operations."

Allowing us to write 5x³ + 3x² + 4x + 1, the very compact and modern notation, and everyone knows that the exponent applies to the x (and not the coefficient) and that you multiply each term before you add them together.