Is it valid to calculate concentrations given equilibrium constant and initial concentration?
The equation
$$\ce{2A &lt;=&gt; B + C}$$
implies a fixed stoichiometry between B and C. If the two separate reactions (2) and (3) were happening, the correct net equation would be
$$\ce{(x + y) A &lt;=&gt; x B + y C}$$
Here is an example that would be well-described by reaction (1):
$$\ce{2 H2O &lt;=&gt; H3O+ + OH-}$$
One water molecule cannot make a hydronium ion (goes against the conservation of mass and charge). The same goes for a water reacting to form a hydroxide. Instead, for every hydronium that forms, exactly one hydroxide is formed. So the separate reactions do not exist, and the separate equilibrium constants do not either.
Of course, there are other reactions where two independent products exist, such as linear glucose forming the alpha or beta anomer of the pyranose ring. In this case, there are two equilibrium constants, but there is no combined equilibrium constant because the two reactions are not the steps of an overall reaction, but rather two distinct reaction paths.