To me, the “is 1 prime” question gets at a very subtle distinction people don’t realize they’re making.
Some people conceptualize the prime numbers as a mathematical sequence definitionally — akin to the Fibonacci sequence. And mathematical sequences have both inductive-case members — that is, members created by the inductive-step “algorithm” for the sequence — and also base-case members — members by fiat, that bootstrap the sequence into being.
In the mathematical inductive sequence of the prime numbers, you clearly need 1 there in order to “get the sequence going.” Any parsimonious formula you could define for a prime-sieve induction step, needs to start off with a 1 already “in the sieve” in order to generate the sequence you want it to generate. Any special-case that excludes 1 (or rather, unit values) from “being prime”, can be reformulated into a syllogism for 1 (/unit values) being base-case members of the primes sequence.
And yet, it turns out that the definition of “prime” isn’t actually “a member of the generative sequence of the prime numbers”, but rather more specifically “an output of the generative process of the prime numbers.”
This is unusual for a mathematical property — usually we use an adjective “Foo x” to refer to an x that shows up as a member in a mathematical object Foo. But prime-ness and membership-in-the-primes-sequence aren’t equivalent. Primality as a property, was created as a mathematical tool to reason about perfect numbers; and you don’t get perfect numbers (or at least, not the same set of perfect numbers) if you label 1 as prime. But this, obviously, has nothing to do with the primes sequence!
It’s as if we had a concept of a “Fibonacci number” that didn’t include the base-case members of 0 and 1. It would feels kind of strange and arbitrary to laymen, who probably know about the Fibonacci sequence, but who wouldn’t realize that there might be something interesting about the Fibonacci-ness property of the inductive members, that doesn’t apply to the base-case members.