Deriving the formula of oscillation frequency for the Phase Shift Oscillator
Your 2nd and 3rd equations are incorrect.

The 1st equation is correct but the 2nd equation should be

$$V_2 = V_1\frac{\frac{1}{sC_2}||(R_3 + \frac{1}{sC_3})}{R_2 + \frac{1}{sC_2}||(R_3 + \frac{1}{sC_3})}$$

In other words, you didn't take into account the loading of the succeeding stages.



According to my morning algebra exercise, for uniform resistor values \$R\$ and capacitor values \$C\$,

$$\frac{V_+}{V_i} = \frac{1}{1 + 6sRC + 5(sRC)^2 + (sRC)^3} = \frac{1}{[1 - 5(\omega RC)^2] + j[6\omega RC - (\omega RC)^3]}$$

The phase shift is \$180^{\circ}\$ when the imaginary part of the denominator vanishes thus,

$$6\omega_0 RC = (\omega_0 RC)^3 \rightarrow \omega_0 = \frac{\sqrt{6}}{RC}$$

For the chosen resistor and capacitor values, the frequency is

$$f_0 =  \frac{\sqrt{6}}{2\pi \cdot 1k\Omega \cdot 100nF} = 3.898kHz$$

To verify this calculation, I simulated the phase shift network and plotted the transfer function: