I disagree with your claim that "[the set of integers] is bigger than [the set of even numbers]." They have the same cardinality, and I'm not sure how else you can reasonably define "bigger" or "size" for infinite sets. If you prefer to treat it like real money, you can buy just as much with {1, 2, 3,...} dollars, and {21, 22, 23, ...} dollars (after you lost $20). Or, from another direction, you could go to a currency exchange and do a real life bijection, trading 20 -> 1, 21 -> 2, 23 -> 3, and so on. Either way, it makes sense to say you have the same amount of money. "Makes sense," since we're talking about an abstraction that doesn't exist.

If mathematically, the sets are the same size, and practically, in terms of buying power, the bankrolls are the same size, I'm not sure how we can reach an abstraction in either perspective where one is "less" than the other.

(The question of walking away is sort of ill-formed for the original Martingale discussion -- since the point there was that you'd stay at the table until you won a spin.)

Interestingly this brings up another simple flaw in the "infinite resources" caveat. When you do finally win a spin, you'll still have aleph-null dollars -- you won't have actually won anything. So the only condition under which the martingale works -- when you can't lose -- is one under which you can't win either. Unless your goal is not to increase your purchasing/gaming power but to break the casino, as in an "Ocean's N" film.