What is the reason behind the current Order of Operations? (PEMDAS)

As Zach Stone pointed out, the order of operations is just a convention, and if you choose to change the order, all that would happen is you would need to use parentheses in different places. Everything would work out fine as long as you made the correct adjustments. That being said, there is a reason for the convention. In some sense multiplication is just repeated addition. Furthermore exponentiation is just repeated multiplication(as long as we restrict ourselves to integers) therefore it makes sense to first turn all exponents into multiplication, then turn all multiplication into addition, and then compute the addition problem. Thus, at least as far as the integers are concerned, there is a natural ordering of the operations based on their definition. It gets more complicated when you start dealing with all real numbers, but the order is inherited from integer arithmetic.

Let's pretend multiplication came after addition. Try writing this without parentheses: $$(a\times b)+c$$ You'll find this very hard. With the current order of operations, we can almost always get rid of parentheses, by using the distributivity laws.

In fact, distributivity is what determines the order of operations. Exponents distribute over multiplication (i.e. $(a\times b)^c=a^c\times b^c$), so exponents come before multiplication. Multiplication distributes over addition (i.e. $(a+b)\times c=a\times c+b\times c$), so multiplication comes first. With PEMDAS, we can get rid of parentheses using distributivity. With a different order ("PEASMD"?), we can't.

In the minds of men and demons alike, multiplication is a more important operation than addition. Also, humans developed algebraic notation from their main language, which was Latin when I was born but is now English.

I still remember when Lincoln gave the Gettysburg address. "Fourscore and seven years ago..." That's $4 \times 20 + 7$ years. In German he might have said "Sieben und achtzig Jahre ­ſind verfloſſen, ſeit unſere Väter auf dieſem Continent einer neue Nation..." That's $7 + 8 \times 10$ years. Dozens, scores, hundreds, grosses, that's how you address a lot of integers, and then you adjust with little additions or subtractions.

A lot of this is just commonsense. But it was the development of computers that forced humans to codify a lot of this commonsense. Whereas a human understands what is meant by $$\prod_{n = 1}^\infty 1 - \frac{(-1)^n}{\phi^{2n}}$$ a computer needs to be given this as $$\prod_{n = 1}^\infty \left(1 - \frac{(-1)^n}{\phi^{2n}}\right).$$

Look into almost any book about a computer programming language, or even a scripting language like Javascript, and one of the very first few things you will see will invariably be a table of operator precedence. A computer, in its glorious stupidity, needs to be told in what order to do operations.

Polynomials are important in their own right, regardless of notation. We have settled on a notation that makes them easy to write. We 'could' just use parenthesis between every operation, but that would be terrible. It's just a convention to simplify reading and writing. Changing the convention wouldn't break anything, we'd just need a lot of parenthesis to express what we want.

One place where polynomials occur organically is in field extensions. One could actually do quite a bit of Galois theory without ever explicitly writing down a polynomial. Similarly, polynomials occur organically in linear algebra (one can use tensor products to abstract abstract away things like characteristic polynomials). It would be harder, but equally powerful. But this helps to justify why we care about polynomials and why we might want to write them down.