I believe I can independently replicate your analysis using freely available SymPy:
z3 = par( R, 1/s/C ) # last dividerz2 = par( z3 + R, 1/s/C ) # middle dividerz1 = par( z2 + R, 1/s/C ) # first dividertf = simplify(1/(1+R/z1)/(1+R/z2)/(1+R/z3)) # all the dividers combinedsolve( Eq( im(fraction(tf)[1].subs(s,I*omega)), 0 ), omega ) # solve imaginary part[0, -sqrt(10)/(C*R), sqrt(10)/(C*R)] # last array item is positive omegaabs( tf.subs( s, I*sqrt(10)/R/C ) ) # magnitude of transfer function1/56 # resulting attenuationI do get the same attenuation as you did.
Picking \$R=2.2\:\text{k}\Omega\$ and \$C=100\:\text{nF}\$, then \$\omega=\frac{\sqrt{10}}{2.2\:\text{k}\Omega\,\cdot\,100\:\text{nF}}\$ or \$f\approx 2288\:\text{Hz}\$.
If I use the exact voltage gain magnitude as derived from theoretical calculations, but inverted to get \$180^\circ\$, then I may expect either gradual damping out or else just slight clipping of the opamp output. However, the output of the phase shift passive stage (and input to the opamp stage) should look very much sinusoidal. So I'll select that.
Plugging into LTspice and using the LT1800 (nice rail to rail opamp) then I get the following:
And I compute \$f=\frac1{120.70785\:\text{ms}-120.269\:\text{ms}}\approx 2279\:\text{Hz}\$. Given the expected clipping effect, that's validation of theory.
(It does take some time to come up. That's why I set it to wait out a few hundred milliseconds to stabilize before presenting the output.)