Conceptualizing Multiple Equilibria
When there is a situation of multiple equilibria, all of them must be satisfied. If this is the case, the whole system will be at equilibrium. 

For your example and as you report, the maths let you write that

$
K_{3}=K_{1}\times K_{2} \quad (1)
$

This final equation, beyond being the mere mathematical combination of two initial equations, is obtained by considering that Gibbs free energy is a function of state: iIt describes "quantitatively an equilibrium state of a thermodynamic system, irrespective of how the system arrived in that state" (Wikipedia). One may get to equilibrium via an unique straight step, or through several steps. But the final destination will be the same.

This definition allows us to write:

$
\Delta G_{3}^{o}=\Delta G_{1}^{o} + \Delta G_{2}^{o}  \quad (2)
$

Equation (2) can be converted into equation (1), by remembering that $
\Delta G^{o}=-RT\text{ln} K$.

We can first of all say that $K_{1}$ and $K_{2}$ relate to the two "pieces" of the reaction path one has to cover to reach the overall equilibrium. Knowing them is essential, if one wants to obtain the quantities for all the chemical species, at equilibrium.

$K_{3}$, on the other hand, relates to the "shortcut" (the unique "ideal" step connecting the initial and final states of a system) and is linked to $K_{1}$ and $K_{2}$, through equation (1).

Using the examples you provide and assuming initially 1 mole for $A$ and 1 mole for $B$, we write:

$
K_{1}=\frac{(y-z)y}{(1-y)^2}
$

$
K_{2}=\frac{z}{y-z}
$

We could more generally write:

$
K_{1}=f(y,z)
$

$
K_{2}=g(y,z)
$

The two constants are different functions: but they both depend on the same parameters, $y$ and $z$, which are temperature and pressure (for gases) dependent. If temperature changes, both constants will accordingly change.

To determine the equilibrium quantities, at given temperature and pressure, we'll need the values for $K_{1}$ and $K_{2}$, since knowing only $K_{3}$ won't be enough.

Beyond all this, kinetics will affect the time needed to reach system equilibrium: but that's another story.