In Organic Chemistry by Clayden, the stereoselectivity and regioselectivity of the E1 mechanism is explained in terms of the differing transition state energies (and therefore rates) of the possible deprotonation steps.
However, the rate-determining step of the E1 mechanism is the formation of the carbocation, not the deprotonation. How is it that the deprotonation is able to control the stereoselectivity and regioselectivity of the reaction despite not controlling the overall rate of reaction?
I have a potential explanation for this, though I am not sure if it is correct. I think that the rate-determining step only limits the rate of the overall elimination, and has no impact on the stereoselectivity. Although carbocation deprotonation has to occur much faster than the carbocation formation, when comparing the two deprotonation routes, one route is still much faster than the other (even though both are quite fast compared to the carbocation formation), which means that the deprotonation is stereo and regioselective, because the majority of the carbocations will react via the faster route. Is this thinking correct?