One reason to not have 1 be prime, is otherwise the prime factorization of natural numbers would not be unique. This property of unique prime factorization is

One reason to not have 1 be prime, is otherwise the prime factorization of natural numbers would not be unique. This property of unique prime factorization is not simply "convenient", it is used to derive a large number of other important results. Do remember that math is a bunch of rules that are made up by people to capture certain abstract concepts, and to understand patterns and relationships between those abstract concepts. So although it might feel like a "kludge" to you, that's just the way math is. You can make up whatever rules you like, and some rules exclude "inconvenient results", leading to a deeper understanding, and thus are more widely adopted by other mathematicians.