Alkene reaction with acid via SN1 reaction
My suspicion is that the resonance structures accessible to the allylic carbocation provide a lower energy state than the formation of the C--H single bond and the isolated carbocation that your expected pathway would require. 

In order of reactivities, you are right to think that the alcohol is the controlling factor because of its pKa. Under acidic conditions it will behave more as a base than the alkene (It is much more basic than an alkene, or think of the Oxygen in an alcohol as being generally a stronger nucleophile than the alkene). This was one main consideration my O-Chem professor stressed was for us to be aware of how other functionalities in our molecules would behave under the reaction conditions of other steps (Eg, if going into a strongly acidic/basic environment after installing a reactive group). I am not an organic chemist by training, but I suspect that if you wanted to access the alkene reactivity, you would want some kind of alcohol protecting group.

This page has a brief discussion about three classes of nucleophile (and the next page describes what makes a good nucleophile): https://www.masterorganicchemistry.com/2011/03/04/the-three-classes-of-nucleophiles/