Break feedback loop of oscillator
When you "break the loop", you're not expecting the new (hypothetial) circuit to still be an oscillator. You're just making clear what the loop path is so that you can see what \$\beta\$ and \$A\$ are. 

In order to see these, you typically inject a new source at the point where you broke the loop.

So your new circuit with the loop broken should look like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Now you can calculate the transfer function from V1 to Vfb to get \$\beta\$, and of course the forward gain is just -K.

In cases where you are modeling the circuit at the level of transistors, you will likely also have to add additional components to maintain the bias point of the circuit, while breaking the feedback loop at the signal frequency.