This issue seems like partly an artifact of the invented binary operator ^. In math exponents are superscripts and there's no binary operator- it's part of the term. But for text on computers, binary operators ended up being fabricated for the things you couldn't represent directly. The caret is common and was the first one to appear, but doubled multiplication signs is another one. Traditionally, binary operators have lower precedence than unary operators since the unary minus is considered part of the term, so the exponent turning into an operator mucks things up if it's still implemented to adhere to that logic.

You sort of see the same issue with division. The forward slash is a completely invented binary operator since the actual division symbol was often not present- and let's be real nobody uses the binary division operator when writing formulae. It's supposed to represent the dividing line in a fraction, similar to how division is usually represented in a formula as a fraction of two other expressions. It's got lower precedence than anything in either term- but, if you just replace the dividing line with a forward slash to input the formula into a computer, you'll get incorrect results, because it's replacing what is part of a complete term (the division line) with a new binary operator inserted between sets of terms, which is now subject to precedence rules.