I think "in the long run" is used in the same sense as for the law of large numbers. The reason we get a different result is that the results of a bet constrain the possible choices for future bets, and it basically turns out that bets are roughly multiplicative in nature, hence why you want to maximize something like log(x) (because if x is multiplicative, log(x) would be additive and law of large numbers applies; that's not a proof but it's intuition).