The same story "could" be told without ever mentioning the OR, like so: as soon as we have that formula_20 and formula_21 then we have that formula_27 Thus if, by random sampling, we manage to estimate formula_24 then, by rare disease assumption, that will be a good estimate of formula_29 which is all we need (besides formula_23 which we presumably already know by studying the few cases of the disease) to compute the RR.
The same story "could" be told without ever mentioning the OR, like so: as soon as we have that formula_20 and formula_21 then we have that formula_27 Thus if, by random sampling, we manage to estimate formula_24 then, by rare disease assumption, that will be a good estimate of formula_29 which is all we need (besides formula_23 which we presumably already know by studying the few cases of the disease) to compute the RR.