Reaction quotient and Gibbs free energy at the start of a reaction
Your interpretation of the equation is correct. $\Delta G^o$ gives you thermodynamic favorability of a reaction under standard conditions (Q=1) and even reactions with positive values of $\Delta G^o$ (unfavored under standard conditions) can be driven to proceed if the concentrations of the reactants and products are made extreme enough. 

The best way to think about the process is via a curve of G vs reaction extent $\xi$, this is related to Q and ranges from 0 (pure reactants, Q = 0, #1 on the graph) to 1 (pure products, Q = ∞, # 2 on the graph):



https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_Chem1_(Lower)/15%3A_Thermodynamics_of_Chemical_Equilibria/23.4%3A_Free_Energy_and_the_Gibbs_Function

Any reaction will have an equilibrium point on this curve, corresponding to a most stable composition (# 3 on the graph). The value of the equilibrium constant (K) tells us the x coordinate of the most stable position (i.e. how much reactant vs product we have in the equilibrium mixture).

No matter what the value of K, it will always be in the range $0&lt;K&lt;∞$. If we start the process with pure reactants, $Q = 0$ and we are to the left of the equilibrium point. The system will shift to the right and some of the reactants will convert into products.

This is what you mean by


  ultimately we'll have ∆G to be a large, negative value which makes the reaction spontaneous?


you can calculate a negative $\Delta G{_r}$ for the reaction when you have no (or infinitesimal amounts of) products.

Many texts are moving away from the term "spontaneous" for this situation, because the system may not shift to the right (reactants convert to products) if there is a kinetic barrier to prevent this, so the forward reaction may not actually be observed. The term "thermodynamically favored" is often used instead, meaning having a negative $\Delta G{_r}$ under these conditions.

You can find lots more about these concepts at the following posts:

If change in free energy (G) is positive, how do those reactions still occur?

Calculating ΔG at the extremes of reaction extent