The set-theoretic definition of infinity came about to make thinking about small numbers easier. As trivial examples, if we reject this definition of cardinali

The set-theoretic definition of infinity came about to make thinking about small numbers easier. As trivial examples, if we reject this definition of cardinality, we can no longer speak of the natural numbers as a set (nor therefore even finite subsets of them), we would lose key set-theoretic definitions of both natural and real numbers, we could not talk set-theoretically about the domain/range of most interesting functions, etc. Honestly, that would be a much more confusing mathematical world than one with Hilbert's hotel or Banach-Tarski spheres.