Michaelis-Menten Kinetics: Does measuring apparent Km and Vmax take into account competing reactions?
Have you heard of something known as "Occam's Razor" ? 

It says when you have multiple possible explanations/hypotheses then select the one which is simplest (i.e least number of assumptions)

Same with mathematical models. Chemical kinetics models usually assume first order unless there is some evidence against it. Similarly, for enzyme kinetics, as long as you don't think there is something anomalous going on, use Michaelis-Menten model. 


  What if there are entire reactions going on that are consuming the
  product or substrate that are not known


Many physiological enzyme catalysed reaction are not Michaelis-Menten. If you happen to have an insight about it or if the simulations dont match the experimental data, you can revise the model. 

However, Michaelis-Menten model for for competitive inhibition exists, as you yourself indicate in the link.