background
I read in the book Introduction to Electrodynamics by D. J, Griffiths the process of solving the four Maxwell equations in the most general form:
Firstly, the four equations were simplified using potentials as
$$\nabla ^2 \vec A-\mu_0 \varepsilon_0 \frac{\partial^2 \vec A} {\partial t^2} - \nabla \left({\nabla \cdot \vec A + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} }\right)=- \mu_0\vec J_c $$
$$\nabla^2 V +\frac{\partial{\left(\nabla\cdot \vec A\right)} } {\partial t} = - \frac{\rho} {\varepsilon_0} $$
Then the Lorenz gauge condition
$$\nabla \cdot \vec A = - \mu_0 \varepsilon_0 \frac{\partial V}{\partial t}\tag1$$
was used saying that this does not change the solution.
I understand that the only condition on $\vec A$ is that $\vec \nabla \times\vec A$ should give $\vec B$ and hence, $\vec \nabla \cdot \vec A$ could be any function and (1) is a good candidate.Also I see how this transformation makes the two above equations symmetric and easier.
Problem
But it seems to me that (1) imposes a new constraint on the value of $V$ at any point. It seems that eq (1) partially determines $\vec A(x, y, z, t) $ at a point from $V(x, y, z, t)$ and vice versa $\left( \text{just like }\ \vec \nabla \times \vec E = - \frac{\partial B}{\partial t}\right) $.
Since all of Maxwell's equations are used up, this new relationship between $A$ and $V$ seems to be a new important relation. This new relation is imposing a new constraint on possible values of $A$ and $V$ apart from the two already present.
Question
Is (1) a completely new constrain which $A$ and $V$ follows apart from the above two equations?
If yes, then why shouldn't we consider it as another Maxwell’s equation and provide a strong proof for this relation rather than calling it a transformation that gives same solutions?
If no, then it should not depend on whether we do Lorenz gauge or not - we should get the answer just from the other two equations. Is it possible?
Note: If the added function (to $A$) was just some $f(x, y, z, t)$ with proper gauge conditions, I would not have this doubt. But here the gauge condition is more of a relation between $A$ and $V$. I'm finding it hard to understand.