Thermodynamics effect of change of temperature on equilibrium
First of all, spontaneity of a reaction depends on the kinetics, not thermodynamics. 

Second, I always imagine LeChatelier's principle to work like this: When you raise the temperature for an endothermic reaction, it has more temperature to work with so the equilibrium shifts to the right. And if you lower the temperature for an exothermic reaction, that means there's more room for the energy created by the reaction.

But if you want to use $\Delta G = \Delta H - T\Delta S$ to check how the equilibrium behaves, you will only see the effect the temperature has on the free energy. It relates to the equilibrium via $$ \Delta G = -RT\ln K \Longleftrightarrow K = \exp\left(-\frac{\Delta G}{RT}\right)$$
This means that if you raise the temperature the expression in the exponent gets closer to 0, which in turn means that the overall $K$ becomes larger. So the equilibrium shifts towards the products. 

However, I have neglected the influence of the entropy above. The formula should be, when completely written out:
$$ K = \exp\left(-\frac{\Delta H - T\Delta S}{RT}\right) $$

And with this you can always see whereto the equilibrium will shift.