Linear Oscillator Frequency Problem
Some of your gain calculations are wrong: should contain terms: R/(1+RCs), 2R/(1+2RCs), 1/2RCs and, of course, gm

Derive the closed loop transfer function, which is 3rd order. For steady-state oscillation the denominator of the CLTF must factorise to the form: (s^2 + wn^2)(s + a), where wn is the resonant frequency in rad/sec. The transient, e^-at, decays to zero, leaving the steady-state sinusoid at wn rad/sec. Compare TF denominator coefficients to find the values of wn and gm

Rough calculation gives: wn = 1/RCsqrt(2); gm = sqrt(3/2)/R, but these need checking.

The above is based on the observation that the denominator of the closed loop TF of a 3rd order system can be written: (As^3 + Bs^2 + Cs + D), which can be simplified by dividing throughout by A, giving the form (s^3 + Ps^2 + Qs + R). Note that the denominator of the CLTF represents the characteristic equation; the numerator only serves to augment the natural behaviour.

The relative stability of a 3rd order CLTF can be determined by comparing the products of the inner two coefficients and the outer two coefficients. Thus, BC>AD = stable; BC=AD = critically stable (i.e. sinusoidal); BC &lt; AD = unstable. This is a pretty useful design ROT for 3rd order systems generally.

So, taking the simplified TF and setting the sinusoidal condition, PQ=R, gives the denominator TF: (s^3 + Ps^2 + Qs + PQ) and this factorises to (s^2 + Q)(s + P), which immediately gives the resonant frequency as wn=sqrt(Q) rad/sec