I don't think I am putting out something truly novel here either; the answer by @MAFIA36790 and the discussion in the link you yourself post it seem quite satisfactory to me, yet I shall try to put a different spin on things so to speak.
Let me define the chemical potential, as the partial molar Gibbs energy of species "$j$":
$$\mu_j = \left ( \frac{\partial G}{\partial n_j} \right)_{p,T,n'}$$
and for a system of components A, B, etc. the fundamental equation of chemical thermodynamics gives us:
$$\mathrm dG= V~\mathrm dp-S ~\mathrm dT+\Sigma \mu_j~\mathrm dn_j$$
Now, I consider a simple equilibria $$R \rightleftharpoons P $$
and additionally, I introduce a parameter " $\xi$ " called the extent of the reaction (it has the dimensions of amount of substance)
for macroscopic changes, the amount of a component $j$ changes by $\nu_j\Delta\xi$ ($\nu_j$ are the "signed" stoichiometric coefficients; positive for products, and negative for reactants)
Now, if we defined $$\Delta _rG = \left ( \frac{\partial G}{\partial \xi} \right)_{p,T} $$
This $\Delta _r G$ still holds the meaning you might already be familiar with, namely:
$$\Delta _r G = \Delta _f G\textrm{ (products)} - \Delta _f G\textrm{ (reactants)}$$
and I have nearly extended it to mean the derivative defined above so that i can relate it to the extent of a reaction (measured in amounts of substance consumed/produced)
and at constant temperature and pressure, from the second equation in this post:
$$ \mathrm dG = (\mu_P - \mu_R)~ \mathrm d\xi$$
so, $$ \Delta _rG = \left ( \frac{\partial G}{\partial \xi} \right)_{p,T} = (\mu_P - \mu_R)$$
Now, as already mentioned, if we were to plot the reaction Gibbs energy as a function of the extent of the reaction, a minimum on the plot would correspond to the first derivative defined above being equal to zero. (or, equivalently in the chemical potential of reactants and products being exactly equal). This is the condition for equilibrium.
Perhaps, stooping down to layman terminology (caution), one can say $\Delta _r G $ is like a "force" that drives a reaction forward, and the system is in equilibrium when this driving force is zero (and moves in the reverse direction when it changes sign).
An appropriate mechanical analogy could perhaps be two weights on either end of a string wrapped across a pulley. If the heavier weight is higher, it would descend raising the lower weight. The system would come to rest when they are in balance.