SN1 Polar Protic Solvent Stabilization of Substrate
I know that the standard textbook answer is that a polar protic
  solvent is able to better stabilize/solvate carbocations through
  hydrogen bonding.


In an $\ce{S_{N}1}$ reaction, a polar protic solvent can stabilize the leaving group through both hydrogen bonding and solvent dielectric effects.  While these same solvent dielectric effects can also stabilize the carbocation intermediate, I don't think hydrogen bonding plays a role in the stabilization of the carbocation itself.  First off, any protic molecule close enough to the carbocation to hydrogen bond would simply react with it instead of forming a hydrogen bond. Second, since the carbocation is positively charged it cannot form a hydrogen bond.  Again, solvent stabilization of the carbocation occurs through dielectric (dipolar) effects of the medium.