I looked this up and apparently there are some old texts on factorials that are written as n * n - 1 * n - 2 etc. from before PEMDAS was standard, and the reader was supposed to add or subtract before multiplying. So it's perfectly acceptable to do math without PEMDAS, you just have to make it clear to the reader, that's the important thing.
In fact, elementary schools could completely eliminate teaching PEMDAS, and just tell students to put parentheses to indicate which operations should be done first. However, the reason we multiply before adding is not because it's mathematically necessary, but because it makes writing polynomials easier. ax^2 + bx + c requires parentheses around the first two terms unless you have a convention that says you multiply before adding.
Another thing is that a + b * c is ambiguous, not unambiguous as some in the thread posted, since the definitions of addition and multiplication don't tell you what order to do those operations in. To create clarity, someone actually has to create a standard of whether a + b * c means (a+b)*c or a+(b*c). That way, problems without complete use of parentheses, like the one in the 80+ page thread, still have one correct solution (even if not everyone can get it right).