It came up a bit in some physics classes, when you can mathematically make something go to positive or negative infinity being able to remove it from the simplified calculation of something is very handy.

On the one hand I get why that is - the calculus notion of infinity is the one that tends to be useful in applied math - on the other hand it's a shame because the set theoretic notion of infinity has more to offer to someone trying to ponder the nature of the infinite.

Or put another way, "what's ∞ + 1" basically invites the non-answer "that's not a well-formed question" whereas "what's ω + 1" gives you a whole intellectual thread to pull on.

I've always been disappointed that number theory, set theory, etc aren't introduced in middle school or high school.

It makes sense, since those are a lot less useful than the subjects that are taught, but something like number theory is incredibly approachable to a middle school student. And it can show students that math can be a lot less about memorization and a lot more about creative thinking w.r.t. proofs.

I would argue that "that's not a well-formed question" is a correct answer, not a non-answer.

...and that the intellectual thread you are pulling on is a (more) artificial notion, constructed by set theorists for the sake of set theorists, not for the sake of counting or measuring in any real sense.

I’d disagree personally. The idea that you can add things to an infinite set, or multiply an infinite set are actually useful concepts. If you imagine the universe is infinite and as such has infinite stars in it (ω) you could discuss how some infinite universes have twice as much star density as our infinite universe (ω2). Or imagine taking a copy of our universe and adding a single star 10 light years from earth (ω+1). In a very real sense the second universe would have twice as much stuff in it as the first universe, even if you can countably map the two universes to each other.

Or maybe put another way, taking the idea that infinity is just infinity makes a lot of sense when you’re primarily considering non-infinite numbers. When you’re primarily considering the concept of infinity and what you can do with it mathematically though, using systems that let you describe infinity with more nuance makes a lot of sense.