When talking about bycicles, it is usual to measure their gears by the number of teeth they have. This measure is then used to calculate the gear ratio: the ratio between the number of teeth of the front gear and those of the back gear. This in turn is calculated for each front-back gear combination available on a bike to describe the range of its achievable gains.
My physics knowledge is very limited, but when I imagine what determines those gains I come to the same conclusions as this answer: the gain is determined by the radiuses of the crank, the chain wheel (or front gear), the rear wheel, and the sprocket (or rear gear).
Of course, if the distance between teeth has to be the same for all gears (due to the chain being the same) more teeth means bigger radius and viceversa.
However, few ever talk about radiuses, most discuss only about the number of teeth. This often leads some to some claim that the number of teeth plays a significant role, thus that different numbers of teeth would lead to different gains despite the radiuses being the same.
This feels odd to me: I have the feeling that having more teeth merely serves to distribute forces among them to prevent them from wearing out, but I'm unable to confirm it because I don't know how to approach the problem from a physics standpoint.
Does the number of teeth (on fixed radiuses) actually matter? If so, what are the principles behind it?