Why don't the solute molecules in a volatile solvent also escape to contribute to vapor pressure?
It is true that "non-volatile" is a relative term and that virtually all compounds have some, possibly infinitesimal, vapor pressure, but the idea is that this is negligible. And you are also correct that "...the molecules of the solute that are at the surface between the solution and the vapor and should not be held by the strong intermolecular bonds of the solid."  

Concerning that latter statement however, there are many reasons that a compound might maintain very low volatility as a dilute solute, even though the intermolecular forces within the bulk solid are not at play. For example, compounds having a large molecular weight tend to be non-volatile, and these will still require much more kinetic energy to be "bumped out" of the solvent, meaning this will be a statistically rare event compared to the transfer of solvent molecules to the vapor phase.  

Ionic solids are another type of solute that are typically quite non-volatile. In order for an ionic species occupying a solute surface site to escape into the gas phase, in the case of an aqueous solvent for example, a rather large amount of energy would be required to dissociate the ion from it's solvation sphere and kick it out into the gas phase as a free ion.  

I hope this answers your question "where did I go wrong in my reasoning". Don't hesitate to ask for clarification in the comments if I missed something.