Differential Amplifier with Hard Limiting
Not an answer, but too big for a comment.

My only confusion is why did he change \$\omega_c \text{ to }\omega_0\$

I would consider only the phase modulation of the peak shift and not the zero crossing.  This is not a limiter but some soft limiting is occurring if the A is say a hard peak limiting value. So the peaks are shifted early and late by PM not FM. So there are alias sum and difference frequencies but still linear gain and linear phase modulation above A but in opposite directions +/-cos for upper and lower alias or difference frequency.

Due to limiting A cannot be increased but the linear amp now multiplies phase for peak excursions of "a" in opposite directions. So the its power is now split to 1/2 and in opposite phase to the difference frequency as upper and lower sidebands due to peak phase shifting  modulation.

Perhaps I am assuming wrong that the limiter action is a hard limiter at inputs above A.

The transfer function of a full limiter is to induce shifting in zero crossings and here the output still has some linear output of say unity gain.

But a hard limiter above some threshold on the output at an equivalent input amplitude A only shifts the peaks. He has rightly ignored  harmonics at 2f, 3f etc , but some energy is lost there, so I would expect sidebands of 1/2a amplitude but not main carrier dropping to 1/2.

Also why is gm and Rd not factoring the ouput levels of A now reduced to A/2 . If it were a hard limiter we can get relative sidebands but not absolute A/2 , a/2 levels out deter ined by input. Perhaps he meant this as a relative level with some gain constant k on all spikes out, then zero-crossing mixer analysis would be done.