As you note, there is no need for operator precedence. There are a number of other conventions that have been used in various contexts. A few programming languages have implemented the left-to-right convention you mention in the question, and a larger number have avoided the problem entirely by using Polish notation or its reverse.
The purpose of operator precedence conventions is to reduce the number of parentheses that are required to unambiguously communicate ideas that occur in practical mathematical usage. The most important rules—that multiplication has a higher precedence than addition, and exponentiation has a higher precedence than multiplication—exist primarily for the convenience of writing polynomials. Without these rules, polynomial expressions would require far more parentheses. Polynomials are so central to mathematical practice that they bleed into the notation, in operator precedence and elsewhere.
Rules for operator precedence beyond the basic arithmetic and relational ($=$, $\leq$, etc.) operators are domain-specific, and authors of papers occasionally have to specify the convention they are using in their notation section. Again, the purpose of having such rules is to improve communication in situations where specifying a convention that the reader has to remember is better than writing parentheses everywhere.
When precedence rules are standardized across an entire mathematical field, it is often because the rules have a clear analogy to arithmetic or relational operators on numbers. For example, concatenation has a higher precedence than union in regular expressions because the regular languages form a semiring, just like all your favorite number systems, and regular expressions are polynomial expressions in this semiring.