There are established theories of probability that have nothing to do with ergodicity. See for example:
Laplace - Théorie analytique des probabilités - 1812
Keynes - A Treatise on Probability - 1921
Jeffreys - Theory of Probability - 1939
Savage - The Foundations of Statistics - 1954
de Finetti - Teoria delle Probabilità - 1970
I can put any probably on anything I want. That doesn’t imply those numbers mean anything usefu or that what I’m doing is backed by any established theory.