It might be helpful if we defined the changes in free energy in these equations in a more precise manner. Let's just consider an ideal gas reaction.
$\Delta G^\circ$ is the change in free energy for the transition between the following two thermodynamic equilibrium states:
State 1: Pure reactants (in separate containers) in stochiometric proportions at 1 atm pressure and temperature T
State 2: Pure products (in separate containers) in corresponding stochiometric proportions at 1 atm pressure and temperature T
To directly measure $\Delta G^\circ$, one would have to dream up a reversible path to transition from state 1 at state 2 and determine $\Delta G^\circ$ for that path. This path might involve the use of constant temperature reservoirs and semipermeable membranes.
$\Delta G$ is the change in free energy for the transition between the following two thermodynamic equilibrium states:
State 1: Pure reactants (in separate containers) in stochiometric proportions at specified pressures and temperature T
State 2: Pure products (in separate containers) in corresponding stochiometric proportions at specified pressures and temperature T
If the specified pressures just happen to correspond to the partial pressures of the gases in a reaction mixture at equilibrium, then $\Delta G = 0$. For small excursions of the partial pressures from the equilibrium values, $\Delta G$ will increase as the squares of the partial pressure increments. That is what we mean when we say that it is at a minimum at equilibrium.
To directly measure $\Delta G$, one would have to dream up a reversible path to transition from state 1 at state 2 and determine $\Delta G$ for that path. This path might involve the use of constant temperature reservoirs, small weights to be added or removed from a piston, and semipermeable membranes.