Why does increasing temperature for an endothermic reaction always increase equilibrium constant?
The first equation, the corresponding change in G,  and the equilibrium constant is supposed to be related to the following two initial and final states:  

State 1:  Pure reactants in separate containers at temperature T and pressure 1 bar

State 2:  Pure products in separate containers at temperature T and pressure 1 bar

The $\Delta G$ you are talking about refers to the change between these two thermodynamic equilibrium states.  But please note that these  thermodynamic equilibrium states that are not in chemical reaction equilibrium with each other.  So, to get the equilibrium constant for the reaction, you are using the $\Delta G$ for these two states which are not in chemical equilibrium.  To understand how this plays out, you need to review the derivation of the relationship between this $\Delta G$ (actually denoted $\Delta G^0$ to distinguish the specific states 1 and 2) and the equilibrium constant K.  

Now, when you change the temperature, all three quantities in the equation for $\Delta G^0$ change:  T, $\Delta H^0$, and $\Delta S^0$, not just T.  This is why the Van't Hopf equation gives the correct result for the effect of temperature on the equilibrium constant, and the other approach you described does not.