A million players each placing a single bet will have an expectation of losing the house edge.
A single player placing a million bets has an expectation of $0.
The fact that the aggregate and the single entity Experience different expectations despite both placing a million bets is what makes this ergodic.
Nassim Taleb also talks about this quite a lot: https://youtu.be/91IOwS0gf3g
TL;DR: while a single investment may be ergodic, portfolio management (the math behind weighting successive and concurrent investments/bets) is not, as it has a strong dependence on all prior states.
Ergodicity is less about memorylessness and more about the constraints on transitions into this or that state. A system is ergodic if "anything that can be an outcome, eventually will happen".