Why do we do open loop analysis for closed loop system? Why cant we directly analyse closed loop system?
Do you know what is needed for a loop to become unstable ?

According to me 2 conditions need to be met for the open loop:

1) the phase has to turn 180 degrees, meaning the signal must invert.

2) the loop must have a gain larger than 1 at the frequency where 1) is valid. Meaning, the signal must be amplified.

Often I can easily determine if these conditions are met for a certain circuit. In practice I perform an AC (small signal) analysis on my circuit, plot the Bode diagrams and I can determine if 1) and 2 ) are met or not.

From the Bode plot I can determine the location (frequency) of the poles and zeros of the open-loop system. When the loop is closed these poles and zeros move according to certain rules. I refer to control theory for this.

Now what would these conditions turn into when the loop is closed ?

Actually I do not know. Also I have no need to know since I open the loop which gives me an insightful way of determining stability.

It is probably possible but much more complex to determine the stability of a closed loop system using poles and zeros.

I can test a closed loop in a different way though. What I then do is simulate the circuit in a transient (time) analysis and when the circuit is in a stable condition (in my simulator) I introduce a current step or voltage step and look at the response. In case of an unstable system I would see that in the response. Even an oscillation might start.