I think the most important reason for keeping infinity around is that the simplest system which describes how numbers work implies the existence of infinitely many numbers. It is much easier to work in this system and then ask "is my answer smaller than 10^50?" than to work in a system using only numbers less than 10^50 and ask "does the result I got actually exist?" at every step.

Similarly, the simplest system that can describe lengths of things is the real numbers. In a way these are make-believe, but it would be ridiculous to try to describe the length of a diagonal of a square without having the concept of the square root of 2.

Comparing the different sizes of infinity is, in its simplest case, a description of how these two systems interact. Don't tell me that's not useful.