Why do collisions in elementary reactions of higher-orders appear to be more likely?
I second J M’s answer: the rate does not depend only on concentrations, but the rate constant is key. However, I will add two more comments:


First, you cannot compare values of rate constants for reactions of different orders… they don't have the same dimension (unit). So, you cannot say things like “rate constants of third order reactions are typically smaller than those of first-order reactions”.
However, you can say (and it is true) that reactions rates of high-order elementary reactions tend to be small, because of the low probability of the molecules crossing path.