Where in those axioms does it say that ↑ = {0|∗} = {0 | {0|0} } is not a number? No where, that's where!

The actual reason that ↑ is simply that it is too ill behaved. The stuff I thought were the "numbers" of combinatorical game are actually just called Conway games. Conway numbers are defined very almost identically to Conway games, but with an added constraint that makes them a much better behaved subset of Conway games.

I suppose you could call this an axiom of combinatorical game theory; but at that point you are essentially just calling every definition an axiom.