Why do we not consider Thevenin Voltage source while dealing with Dependent sources in RLC circuits?
I believe what they are saying is the dependent source and the two resistors can be boiled down into an R-equivalent resistor (with the excitation source removed). This would be advantageous because one could then analyze the circuit as an RLC which simplifies the math.

So what they do is remove the L and the C and subsitute a current source, and then say "Hey, all these components are linear and function like a resistor" 

If you go around the loop and find the equivalent to Vtest you get a resistor, this allows you to ignore 

What they are trying to show is that for some circuits it is easier to simplify the circuit before trying to solve the circuit. 

We are not concerned with the forced response of the circuit because there is no zero state response, the circuit has to be excited for there to be a response.

In addition 1A current through the loop helps you find Vtest for an initial condition, which will help you find Vc, because when the circuit starts up, the inductor is shorted.