Why doesn't the enthalpy change of a reaction affect the rate of reaction?
The rate of a chemical reaction is directly dependant to the activation energy of that particular reaction. These two values are inversely proportional. The higher the activation energy, the lower the rate. As it has been mentioned though, the rate can be increased, in such cases, by increasing the temperature. All of the above is described as kinetics, which shows how fast the reaction will reach its end phase.

Enthalpy is linked with thermodynamics, which shows what kind of entities will be present at the end of the reaction. At what degree the reactants will turn to products.

I don't know your chemical background, so I tried to make this as simple as possible.