Risk-neutral probabilities are not beliefs. They are pricing weights implied by no-arbitrage, and confusing the two is the most common misunderstanding here.
The binomial model shows why: replicate the option with stock and cash, and the price falls out without the real probability of an up move ever appearing. Two traders who disagree violently about direction must still agree on the option price.
The formal statement: discounted asset prices are martingales under the risk-neutral measure.