I saw a question on how to predict the product of a reaction between (hydroxymethyl)cyclobutane and HBr. The correct answer was given as 1-bromocyclopentane, however, I do not understand why a ring expansion would occur in the reaction.
My attempt: I would think that the final product would by (bromomethyl)cyclobutane. This is due to the fact that HBr would first protonate the alcohol, causing it to become a good leaving group. Afterwards, an $\ce{S_N2}$ reaction between this protonated alcohol and $\ce{Br-}$ would take place, as the substrate is a primary alcohol, where no ring expansion/carbocation formation could occur as the steps are concerted, resulting in (bromomethyl)cyclobutane.
Why is this mechanism wrong, and why does a ring expansion occur instead?