Theoretically, it is possible for a gambling site to profit from a 0% house edge gambling game. There are two reasons why, first is bet size limits and the house having more money than the player.
False. The rest of your post with the scenario does not matter.
It doesn't matter whether there are bet limits
or if the house has more money than any given player (or heck, even all the players). If they bet some amount, they have an equal chance of winning and losing. That's it.
Greed doesn't matter. They could still very well win
all their bets or lose
all their bets. Once you get past this, it's easy to understand that with variance, martingale can end up with a player busting but when you consider the entire space of players (we'll call this S), it's different.
The expected value of the amount that S is returned is going to be 1. And this value is equivalent to the amount that the house gets: 1.
If
statistically the house makes no money on each bet, how can you expect to profit regularly?
Hoping that individual players get greedy doesn't help because of the very question: what if they win? There isn't a house edge to make the chance of them losing more likely.