Can hydrogen bonding account for viscosity?
Neither ethanol nor methanol, the alcohols in which hydrogen bonding most strongly operates, are particularly viscous (the same for water). Therefore, viscosity cannot solely be attributed to hydrogen bonding.
On the other hand, when both alcohols and alkanes increase in chain length, they tend to become more viscous and syrupy. This intermolecular force can be attributed to dispersion forces. As the chain length increase there is more possibility of chain/chain interaction, and boiling points increase. In long chain alcohols, the polar heads would then to aggregate, making the chain/chain interaction stronger (cf. detergents).
I don't know if you have ever handled glycerol, 1,2,3-propanetriol. This material is very syrupy. Here, of course, increased hydrogen bonding does operate with respect to say propane or the propanols, and is responsible for the high density (