phase-shift oscillators
My understanding is that the Barkhausen Criterion is a necessary condition for oscillation.

You are correct. It is a necessary condition for sustained, stable oscillations, that neither decay away, nor grow without bound. But,

But this condition isn't sufficient is it?

No it isn't, and you are correct here too. If the closed loop gain is exactly one, at a given frequency, then a small oscillation would stay small, a large oscillation would stay large, and non-oscillation would stay zero!

What additional conditions would guarantee oscillation?

A way to guarantee oscillation is to have a variable gain. The gain should be above 1 when the oscillation amplitude is small, less than 1 when the oscillation amplitude is above a certain level, and exactly 1 when the oscillation is at the target amplitude. There are a number of ways to achieve this, such as taking advantage of non-linearities in certain components, or changes in component values based upon their heating.

Why is the oscillation sinusoidal?

There are a number of points to make here. Because of the necessity of adjusting the gain so that it is below 1 for &quot;too large&quot; oscillations, and above 1 for &quot;too small oscillations&quot;, oscillators generally do not produce perfect sine waves. Often there is a trade off to be made. To guarantee quick start up, one wants the gain to be quite large at low amplitudes. This can mean distortion. Similarly, when the amplitude is too big, and one relies upon non-linear components to cut the gain, the drop in gain often comes about with non-zero distortion. So, the trade off is between the purity of the sine wave, and the stability of the oscillator.
The second point to be made is that the waveform is nearly sinusoidal because sinusoids are the solutions to the differential equations that correspond to the phase shift networks. Or, looked at another way, there is (hopefully) a unique frequency at which the closed loop phase shift is 0. Thus, all frequencies other than that frequency will decay and die out, leaving only a sinusoid.

Can someone point me to a reference that might offer some sort of &quot;middle of the road&quot; answer?

Hopefully someone else see this and knows of a good intermediate level reference.