An odds is the ratio of the probability of an event to its complement:
$$\text{odds}(X) = \frac{P(X)}{1-P(X)}$$
An odds ratio (OR) is the ratio of the odds of an event in one group (say, $A$) versus the odds of an event in another group (say, $B$):
$$\text{OR}(X)_{A\text{ vs }B} = \frac{\frac{P(X|A)}{1-P(X|A)}}{\frac{P(X|B)}{1-P(X|B)}}$$
A probability ratio1 (PR, aka prevalence ratio) is the ratio of the probability of an event in one group ($A$) versus the probability of an event in another group ($B$):
$$\text{PR}(X)_{A\text{ vs }B} = \frac{P(X|A)}{P(X|B)}$$
An incidence proportion can be thought of as pretty similar to a probability (although technically is a rate of probability occurring over time), and we contrast incidence proportions (and incidence densities, for that matter) using relative risks (aka risk ratios, RR), along with other measures like risk differences:
$$\text{RR}_{A\text{ vs }B} = \frac{\text{incidence proportion}(X|A)}{\text{incidence proportion}(X|B)}$$
Why are relative probability contrasts so often represented using relative odds instead of probability ratios, when risk contrasts are represented using relative risks instead of odds ratios (calculated using incidence proportions instead of probabilities)?
My question is foremost about why prefer ORs to PRs, rather than why not use incidence proportions to calculate a quantity like an OR. Edit: I am aware that risks are sometimes contrasted using a risk odds ratio.
1 As near as I can tell… I do not actually encounter this term in my discipline other than very rarely.