Just speaking in general, RR scales will get you into more trouble that OR/HR scales as covariate effects must be restricted in RR scales, e.g., you can’t double risk if the risk is already 0.51.
This also creates a problem in interpreting effect modification
See examples in this paper: Redefining effect modification
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this is a nice paper
I wonder how big a problem this is in the common scenario of common baseline risk and modest intervention effects
I think that has been the problem that perpetuated the use of the RR - with small baseline risks and modest intervention effects the RR numerically may not deviate from the OR but if the two effects are considered equivalent because of their numerical coincidence then it also means that they have been misunderstood.
Note that if there is a common baseline risk and it does not change then all three effect measures will return the same predicted probability under exposure or intervention when modeled in a GLM. The problem starts with changing baseline risk.
Coming back to interactions and random error - there is no way to distinguish (in a single trial) an interaction from an artifact of the sample and majority of interactions are just describing the sample artifact - even when the P for interaction is <0.05. My guess is that the only way to decide if an interaction is not a sample artifact is if it stands across studies e.g. in a meta-analysis