You are talking theory.. (everyone knows that part if you have been in the gambling threads for a good time)
I am talking psychology. Many want quick profits, like to take risks. My theory is they will go on playing till they lose.
Suppose I start a site with -0.1% edge, you make a profit, you will most likely play again. You lose, you think EV is positive and still play again. IT goes on till you lose and have nothing to continue bet. (assuming bankroll of player < house and there is a max bet limit, as I said)
That's the key assumption you're making here, right? For example, if bankroll of player > house, then you will also agree that the house will lose, even if the player wants to keep playing, right?
With that established, we need to see that the players' bankroll is not that of any individual player, but the AGGREGATE of all players. Because what is happening, is the house is essentially gambling against all these players, so the net bankroll they're putting their own against is the total of all players that are playing against them.
Furthermore, the strategy of the aggregate player is simply that for some subset of the bankroll (i.e a single player), a certain playing method is used, i.e martingale or whatever. As we know, the expected return of any strategy is the same. There is no winning/losing strategy.
So I think with those points, we can safely establish that the house will lose. I can understand how you still might feel different; I would say that kind of psychology is akin to the gambler's fallacy. It feels like it makes sense, but in reality it isn't that way.
EDIT: To add, we can use the kelly criterion with variables being the aggregate player bankroll and the ev to calculate exactly how many players need to play in order for it to be "worth it", i.e the players have a good chance of beating the house.