Several comments are in order.
Not being dependent on q is like not depending on the underlying asset expected return. The only random variable over the period important to the option price is the change in the underlying asset price, u or d . But, again, with the binomial model postulated here, any such influence, if there is one, must enter indirectly through the other variables S, u , d , or r . In any model in the social sciences, it is prudent to ask what is being assumed about human behavior and psychology. In this case, we only assume that investors price securities so that there are no riskless arbitrage opportunities.