The Law of Large Numbers is about the average, so it's not relevant here (an average of +1 would mean you got heads every single time, which is extremely unlikely for large n).
If you are looking at the sum, then the value depends on whether the number of trials (n) is even or odd. If n is odd, you would indeed get two peaks at 1 and -1, and you would never get exactly 0. If n is even, you would get a peak at 0 and you would never get exactly 1 or -1.
The expected value (aka average) is a number, not a distribution. The expected value for the sum is 0 even when n is odd and you can't get exactly 0 -- that's just how the expected value works (in the same way that the "expected value" for the number of children in a family can be 2.5 even though a family can't have half a child). If you look at the probability density function for a single trial, then it does have two peaks at 1 and -1 (and is zero everywhere else).
The curve you refer to might be the normal approximation (https://en.wikipedia.org/wiki/Binomial_distribution#Normal_a...). It's true that the normal approximation for the distribution of the sum in your gambling game has a peak at 0 even when n is odd and the sum can't be exactly 0. That's because the normal approximation is a continuous approximation and it doesn't capture the discrete nature of the underlying distribution.